
,-\ 



Class — aAj^. 5i_a2i_ 
Book ^^ Qa 6 

COPyRJGHT DEPOSm 



ELEMENTARY CALCULUS 



A TEXT-BOOK FOR THE USE OF 
STUDENTS IN GENERAL SCIENCE 



BY 



PERCEY F. SMITH, Ph.D. 

PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL 
OF YALE UNIVERSITY 



o>»<o 



NEW YORK-:. CINCINNATI •:• CHICAGO 

AMERICAN BOOK COMPANY 



LiBRaKV -'CONGRESS 
Two Cftpiefe Received 

MAR 17 1904 

Copyrignl trf»try 

CLASS c^ XXd No- 

7 vT i> ^ 

CO^Y S 






Copyright, 1902 and 1903, by 
PERCEY F. SMITH. 



EL. CALC. SMITH. 
W. P. 3 



PREFACE 

This volume has been written in response to the un- 
mistakable and growing demand for a text-book on the 
Calculus which shall present in a course of from thirty- 
five to forty exercises the fundamental notions of this 
branch of mathematics. In American technical schools 
students pursuing courses distinct from engineering 
branches usually terminate their mathematical studies 
with Plane Analytic Geometry. But in view of the recent 
remarkable development of certain of the general sciences 
along mathematical lines, such a course can no longer be 
regarded as adequate. Moreover, there can be no differ- 
ence of opinion as to the relative advantage to the student 
of a knowledge of more than the mere elements of Ana- 
lytic Geometry and an introductory acquaintance with the 
Calculus. It is, I think, the experience of every teacher 
that the average student first realizes the power and use 
of mathematics when taught to solve problems in maxima 
and minima by means of the methods of the Differential 
Calculus. Certainly no stronger argument can be adduced 
in favor of an adjustment of the curriculum which shall 
inclur^e this branch of mathematics. Such a change has 
been effected in the Sheffield Scientific School, and results 
abundantly justify the step. 

For the general student in our colleges who elects a 
year's work in mathematics beyond the usually required 

3 



4 PREFACE 

Trigonometry, the most satisfactory course would seem to 
be one in which the time is equally divided between Plane 
Analytic Geometry and Calculus. 

In writing this book I have everywhere emphasized the 
possibility of applications. The examples have been care- 
fully selected with this end in view. The first chapter 
may seem long, but the notion of limit certainly demands 
adequate treatment. While an elementary text-book offers 
no excuse for employment of the refinements of modern 
rigor, I have endeavored to avoid positive inaccuracies 
and have carefully distinguished between demonstration 
and illustration. 

I am indebted to my colleague, Dr. W. A. Granville, for 
many helpful suggestions. 

PERCEY F. SMITH. 
Sheffield Scientific School, 



CONTENTS 



CHAF 
I. 


PER 

Functions and Limits . » * , 


PAGE 
7 


II. 


Differentiation 


. 23 


III. 


Applications 


. 51 


IV. 


Integration 


. 70 


V. 


Partial Derivatives . . . . 


. . . 84 


VI. 


Additional Examples . . . . 


. 90 



ELEMENTARY CALCULUS 

CHAPTER I 
FUNCTIONS AND LIMITS 

1. Continuous Variation. In this book we are concerned 
with real numbers only. Geometrically, such numbers may 
be conveniently represented by points of a scale (Fig. i). 

. — ' 1 ' j: 1 I 1 i \ 1 1 i 1 — 

etc. -5 — 4 --3 ~3 -1 1 2 8 4 5 6 7 etc. 

Fig. I 

Then to every real number corresponds one point of the 
scale, and only one ; conversely, every point of the scale 
represents a real number. Any segment of the scale, 
however small, represents indefinitely many numbers. We 
speak indifferently of the number a and the point a of the 
scale. 

A variable x is said to vary continuously between the 
numbers a and b when it assumes values corresponding to 
every point of the segment ab. 

2. Functions. The problems arising in Elementary 
Calculus involve in general two variables in such a way 
that the value of one variable can be calculated as soon as 
a value is assumed for the other. Thus, in Geometry, the 
student has an illustration in the area and radius of a 
circle, two variables such that the area A can be calculated 
when we know the radius r from the formula A = irr^, 

7 



8 FUNCTIONS AND LIMITS 

Definition. A variable is said to be a ftcnction of a second 
variable when its value depends upon the value of the sec- 
ond variable and can be calculated when the value of the 
second variable is assigned. 

The first variable is called the dependent variable^ and 
the second the indepeiident variable. 

For example, the equations 

^ = ;r2, J/ = sin;ir, y = \og^^{x^ - l) 

state that j/ is a function of x. In the first two cases, j 
may be calculated for any value of ;r ; in the last case, how- 
ever, X is restricted to values numerically greater than i, 
since the logarithms of negative numbers cannot be found. 
In the first two cases, then, we say that the dependent 
variable (or the function) is defined for every value of ;r, 
and in the last the function is defined only when x exceeds 
I numerically. 

A function is defined for a value of the variable when its 
value can be calculated for that value of the variable. 

Elementary Functions. Power Function: x"^, m any 
positive integer. 

Logarithmic Function: logaXy ^>o; this function is 
defined only for x>o. 

Exponential Function : a"", a>o, i.e. the exponent is a 
variable, the number a being a constant. 

Circular Functions :^ sin;r, cos;r, tan;tr, etc., i.e. involving 
^p the six trigonometric functions. 

* So called from the use of the circle in their definition, 

e.g. in Fig. 2, sin ^(9P= — = MP if R is the unit of 

linear measure. Hereafter, angles will always be meas- 

arc _ AP 
radius R 




ured in circular measure, i.e. x — ^^^ = — . In the 



Fig. 2 nnit circle, Rz:z\, x = arc A P. 



FUNCTIONS AND LIMITS 9 

Inverse CiraUar Functions: arc sin x^ arc tan;ir, etc., i.e. 
the *'arc whose sine is ;r," **arc whose tangent is x^' etc. 
In the unit circle (see Fig. 2) R = \\ if x=MP, then 
arc sin;r= ^XQ AP. 

One thing is peculiar here. Assuming any value of x 
not exceeding i nnnierically , arc sin x may be calculated, 
but the number of answers is always indefinitely great. 
For not only is 

arc sin MP = arc AP, 

but also equal to any number of circumferences + arc AP, 

i.e. divc sin MP = 2irc AP + 2 irn, 

where n is any integer. 

For this reason the inverse circular functions are called 
many-valued functions. For definiteness we may always 
take the least positive arc. 

3. Functional Notation. As general symbols for func- 
tions of variables we use the notation 

fix), e{y), <}>{r), etc., 

(read / function of Xy theta function of y, phi function of 
r, etc.). 

We mean by this that f(x) is a variable whose value de- 
pends upon Xy and can be found when a value is assumed 
for X. The notation is extremely convenient, for it enables 
us to indicate the valine of the fuiiction corresponding to 
any value of the variable for which the function is 
defined. 

Thus /(^) represents the value of /(;r) for x^ay 0(6) 
the value of (y) for y = o, (f){^) the value of <^(r) for 
r^^y etc. 



10 FUNCTIONS AND LIMITS 



EXERCISE 1 



1. For what values of the variable are the following functions 
defined ? 

(a) -. Ans. For every value except x = o, since - cannot be 

^ calculated.* ° 

(d) y/x^ — 6x. Since x'^ ~ 6x or x(x — 6) must not be negative^ 
;f and x — (^ must always have the same signs. 

Ans. For every value except those between o and 6. 



{c) y/y - y'^\ (d) v^io; {e) arc sin r; 



(/) arc sec ;r; {g) sin Vi + x\ (/i) log tan ;r. 

2. Given /(;ir) = ;ir8-7;ir2+i6:r- 12, show that /(2) =o, /(3)=o. 
Does /(^) vanish for any other value of ;r ? 

3. Given /(x) = log;r; show that 

4. Given <^ (x) = a"" ; show that 

cl>(x)cl>(y) = ct>(x + y), 

5. Given 6 (x) = cos;r; 

then 0(x) + 6 (/) = cos x + cos>^. 

From Trigonometry, we know that 

cos X + cos/ = 2 cos i(x-\-y) cos i(x —y) ; 

4. Graph of a Function. After determining for v^hat 
values of the variable a given function is defined, it is im- 
portant to know in what manner the value of the function 

* The student should observe that the four fundamental operations of arithmetic, 
addition, subtraction, multiplication, and division, when performed with r^a/ num- 
bers, give real numbers, with the single exception that division by zero is excluded. 



FUNCTIONS AND LIMITS 11 

changes with the variable. Geometrically this is accom- 
plished by drawing the graph of the function^ which is 
thus defined : 

The graph of a function is the curve passing through all 
points whose abscissas are the values of the variable and 
ordinates the corresponding values of the function. 

In the language of Analytic Geometry the graph 
of a function f{x) is the locus of the equation 



1" 



Fig. 3 

By carefully drawing the graph of a func- 
tion a good idea is obtained of the behavior 
of the function as the variable changes. For ex- 
ample, the graph of loga ;r, i.e, the locus of the 
equation * 

is drawn in Fig. 3. 
/ Here we see the following facts clearly pictured 

to the eye. 

{a) For x=iy logger =log3 i =0. 

(b) For x> ly logger is positive and increases as x in- 
creases. 

* The values of>' are found from the formula proven in the theory of logarithms, 

, ]0g]C)X 



12 



FUNCTIONS AND LIMITS 



{c) For x<i, log^x is negative and increases indefi- 
nitely in numerical value as x diminishes. 

(^) For ;r = o, logg x is not defined, since the logarithm 
of zero cannot be calculated. 

The graph of the general logarithmic function loga^ 
may be drawn by merely changing the ordinates in Fig. 3 
in the constant ratio i -r- logg a. 

Graphs: {a) Oi x\ (J?) Oi xK 







r 










































































































1 


















































































































1 


































/ 


































/ 






























f 




/ 






























3 


/ 

































/ 
































i 


1 






























<. 


7 


















A 


'-■' 















1 


2 ; 


5 










K 




























































( 


a 


) 








































1 



























































_JL_ 




_ 7 




^ t 




A ■ ■ — 




iX 


X' 2 


^0l3j: X 


X 




t - 




/ _ 




r _ 




± _ 




j 


(?>I 


:±i-^ i 


/ 



The graph of r"* has the appearance of {a) or (J?) according as m 
is even or odd. 

(0 Of lOggX. 













































































































































































































9 
















































-1 












^ 


[^ 










" 






























^ 


Ur 




























X' 















/ 


1 


2 




3 


i 




















X 














/ 
















































/ 










































































































































































i 


c 


) 
























































Y' 

Since if we set y = log3 x, then x = 3^, the graph in (c) has the 
same relation to XJC' and KK' as (^) to KK' and XX f. 



FUNCTIONS AND LIMITS 



13 



{e) smx. 



X' 





— t 


— r 




— 


— 


— 




— 
























~ 








~~ 




































"~ 
































_ 














































— 


— 




— 








~ 




























~ 








-ll 






/" 




--- 


Vp 
















H 


s 




-J 








"y 


<i 




J 


i 




^ 


^_ 






_ 


^ 




_ 


T^ 






J 


t 




/ 




-^ 






;j' 


^ 






-X 








-\ 


•^ 


n 


y 








i 




















~" 


... 






































































- 


z 










































~ 


~ 




























_( 


e 






~ 
















































] 


r 







































tan;ir 




























■ 


1 


























































































/ 






























/ 






























f 






















/ 








1 _ 






















/ 








/ 




















/ 










/ 








/ 










-et 


/ 


-'^ 


-1 


0/ 


' I 


2 


s 


/ 








X' 








■{-JL- 


r^y 


i^r i^ 


/ 


IT 






y" 




/ 








i 


ly 




i 


f 












1 












/ 




/ 


























/ 




























I 






























1 






/ 
























1 




, 


























/ 




/ 


























j 












itW 
















l_ 














LL 





F' 

The graphs of the circular functions have the appearance of a curve 
repeated over and over as the variable increases or decreases. As in 
{c) and (^), if we revolve {e) and (/) around XX\ and interchange 
XX^ and KK', we shall have the graphs of arcsin;ir and arc tan ;r 
respectively. 

5. Limits. For the study of the Calculus it is absolutely 
essential that the student should understand perfectly the 
fundamental notion of a limit. He is already familiar with 
simple examples of limits from Geometry, such as the limit 
of the perimeter of an inscribed regular polygon as the 
number of sides is indefinitely increased is the circum- 
ference, and the limit of the area of the polygon is the 
area of the circle. These are examples of variables ap- 
proaching limits, the variable being in the first case the 
perimeter, and in the second the area of the regular 
polygon. The following definition states the matter gen- 
erally. 



Definition. A variable is said to approach a number A 
as a limit when the values of the variable ultimately differ 



14 FUNCTIONS AND LIMITS 

from A by a number whose numerical value is less than 

any assignable positive number. 

If we represent the values of the variable by the infinite 

sequence 

a^ as^^ a^f •••, a^y ^«+b •••> 

then on the scale (Fig. i) the points corresponding to 
^v ^2> ^3> *"' ^ny ^n+b '"y ^tc, will ultimately approach 
nearer the point A than any assignable length, that is, will 
*'heap up" at the point A, The definition interpreted 

A 

-4- 



^1 ttg \^-h- 

Fig. 4 



-h-^ 



geometrically means, then, that no length h (Fig. 4), however 
small, can be laid off from the point A, but that points of 
the sequence will fall within the segment. 

We write Limit (^„) = A, or, also, if we denote the varia- 
ble whose values are a^, a^, etc., by x, 

Limit (;r)= A. 

6. Limiting Value of a Function. Continuous Function. 

Consider the elementary function loga;r (Fig. 3). Take 
any sequence 

^l> ^2> ^3> ***' 

of positive numbers whose limit is some positive number-^. 
For example, the sequence 

1.3, 1.33, 1.333, •-, 

the limit of which is |-. Consider now the sequence of 
numbers 

and draw their ordinates in Fig. 3. Then the student 



FUNCTIONS AND LIMITS 1 5 

will see that this last sequence has the hmit loga(^); 
that is, 

When the variable x approaches a limit A greater than 
zero, the logarithmic function log^x approaches the limit 
logaA, 

We express this important fact by writing 
Limit (loga ^)x=^A = loga A . 

The general relation brought out by this example is the 
following: When the values assumed by the variable x 
approach * a limiting value A, then the corresponding 
values of the function will also approach a Hmiting value ; 
and if the function is defined for the value A, then the 
limiting value of the function is its value for x=^ A. Or, 
in symbols, ii /{A) is a number, then 

Limit (/(^)),=^=/(^). 

For example, since cos o = i, Limit (cos ;r),^o = i- The 
property above described is that of continuitjy ; i,e, a con- 
tinuous function is such that 

Limit fix) = /( Limit x). 

For the purposes of the Calculus it is essential that a 
function should be continuous. The elementary functions 
of § 2 possess this property. 

7. Infinity. If the points on the scale of Fig. i cor- 
responding to the sequence of values of the variable x 

* The variable x may approach the limit A in any manner consistent with the 
definition of the function. In the above illustration the geometrical sequence 

I. I + i. I + i + 1^. I + i + A + a^4. etc., 
whose limit is f, might also have been taken. 



l6 FUNCTIONS AND LIMITS 

ultimately advance to the right without limit, we say, ^^ x 
increases without limit," or also, '' x approaches the limit 
positive infinity," and we write 

Limit ^ = + 00. 

If under the same conditions the points advance to the 
left without limit, we say, **;r decreases without limit," and 
write 

Limit ;r = — 00. 

Finally, if the points advance both to the right and left 
without limit, we write 

Limit ;r =00. 

The student should disabuse his mind of any previous 
notions of infinity not agreeing with the above definitions. 
The symbols + oo, — oo, oo, must be used always in the 
sense above described. 

8. Fundamental Theorems on Limits. The student is 
asked to accept the following theorems as true : 

Given a number of variables whose limits are known ; 
then 

L The limit of an algebraic sum of any finite number 
of variables equals the same algebraic sum of their respective 
limits, 

IL The limit of the product of any fi^iite number of 
variables equals the product of their respective limits, 

in. The limit of a quotient of tzvo variables equals the 
quotient of their respective limits whe7i the limit of the 
denominator is not zero. 



FUNCTIONS AND LIMITS 

9. Two Important Limits. To prove * 

Limit [^]^^^ = 1. 

In Fig. 5 let ;r = arc A T 
= arc A 5, the radius OA 
being taken equal to unity. 
Then 

sin x = MT= SM, 

tan;r= TQ = QS. 

Now 



i; 




Fig. 5 



ST<arcST<SQ + QT. 
.', 2 sin^< 2;r< 2 tan;r; 
whence, dividing through by 2 sin;r, 



T ^ ^ <^ tan;r 
sin X sin x 

Therefore, taking reciprocals, 



cos;ir/ 



^ sm X ^ 
cos;r< < I. 



Now let X approach zero as a limit ; then, since cos o= i, 
sin-T 



and the value of 
have 



X 

Limit 



lies between i and cos^, we must 



sm;r 

X 



= I. 



10. Consider next the infinite sequence 



^' ^+? ^+7^' ' + 



1-2 I • 2 • 3 



♦ Since fnr x — o, 
not defined for x = o. 
EL. CALC. - 



- = -, a meaningless expression, the function 
o 



i8 



FUNCTIONS AND LIMITS 



Representing the successive terms by a^, a^y ^3, •••, we 

have 

^1= I, 



^2 = I + 7> 
^3=1 + 



^4=1 + 



^n= I + 



I -2' 

1 



1-2 I • 2 • 3 



= I 
= 2 

= 2.5 

= 2.666 •- 



+ 



1-2 I • 2 • 3 



^ +••• +.— '-T*, etc. 



;/ — I 



The numbers of this sequence continually increase. We 
may show, however, that any term is less than 3. 
If r > 3, then [r > 2^ and therefore 



I I I I I 2" 

n I 2 2^ 2"^ 2" ^ 



— I 



— I 

2 



smce 



I +- + -4 + -^ + --- +-4=1 

2 2^ 2^ 2" 



is a geometrical progression and its sum may be imme- 
diately written by the usual formula. 

Hence ^n < 3 z]y ^^^ taking ;/ = i, 2, 3, etc. ad infi- 

2^ 

nitunty every term of the sequence is seen to be less than 3. 

e 
cti a2 c^s I 



r 



Fig. 6 



The points, then, corresponding to the sequence (Fig. 6) 
must heap up at some point to the left of 3 ; that is, the 
sequence must have a limit. 

* The symbol \n — i , read " factorial n— i," means the product of all integers 
from I to » — I inclusive. 



FUNCTIONS AND LIMITS 



19 



The calculation of this limit to any number of decimal 
places is a matter of no difficulty, as the following compu- 
tation to five decimal places will show. 



Write down 
Divide by 



i.oooooo(= i). 

2)1.000000 ( = p) 

3). 500000 (^=10 

4). 166667 (=^) 

S).04i667(^=,^J 

6).oo8333 (=|j) 
7).ooi388(=|) 
8).oooi98(=ji) 
9).oooo25 (^=.-|j 
10). 000003 (=1^) 



Adding, 



2.71828 



neglecting the figure in the sixth decimal place, of which 
we cannot be sure. In fact, it can be easily shown that 



20 FUNCTIONS AND LIMITS 

2.71828 is the limit of the sequence correct to five deci- 
mal places. 

Writing the limit of the sequence in the form of an infi- 
nite series and denoting this limit by e, we have 

e = l+i-f — + i + ~ + ,^+ etc., ad infinitum. 

1 [2 [3 [4 [5 

e = 2.71828.... 

The number e is called in the Theory of Logarithms the 
Napierian base or natural base^ and is a number of prime 
importance in mathematics. 

The expression for e in the form of an infinite series 
should be remembered and also its value to five decimal 
places. 

11. To prove 

Limitri4--T = e. 

A rigorous proof of this very important limit is beyond 
the scope of this volume. We may perhaps best illustrate 
the meaning of the theorem by drawing the graph of the 
function for positive values of z. 



then 



Setting ^=^^i+ij, 

logioJ»^ = ^logio(n-^), 



and for any value of z greater than zero y may be approxi- 
mately calculated, as for example in the accompanying 
table, which gives j/ to five decimal places. 



FUNCTIONS AND LIMITS 



21 



Z 
.01 
.1 

I. 

10 

100 

1000 

10,000 

100,000 

1,000,000 



etc. 



y 

1.04723 
1.27098 
2. 

2.59374 
2.70481 
2.71692 
2.71815 
2.71827 
2.71828 




Fig. 7 



The figure illustrates the theorem in showing that the 
graph approaches the line 7 = ^ as z increases indefinitely. 
When z diminishes toward zero, y approaches unity. 



22 FUNCTIONS AND LIMITS 

EXERCISE 2 

[The graph of the functi07i considered nmst be drawn in every case. 2 
1. Prove Limit [" -^^-3^ + 4 1 ^ 2. 

L X—l Jy^g 

We have merely to substitute 2 for x. 

—\ =-2a, 

x^ a Jx=-a 

We cannot substitute directly, for we should get -, a meaningless 
;i'2 - ^2 o 

expression. But = x — a^ and we may now substitute. 

x^ a 

3. Prove the following in which a is any number greater than zero; 
Limit f-^ I = + 00; Limit I-] =00; 

Limit \(i^\ = 00 ; Limit — =0. 

The last three results are often written 

a a 

- = 00, ^ • CO = 00, — = o, 
o 00 

but the student must remember that such equations are merely abbrevia- 
tions of the preceding. 



4. Prove Limit [ ^^-^ 1 = -J^. 

Hint. Multiply numerator and denominator by Vx-\-/i + Vx. 

5. Show that 

Limit r^^l = I ; Limit ftanrl = 00 ; 
Lsin x-Jx=o L -1^^^? 

Limit loge x\ = - 00 ; Limit [^"'1 = o. 



CHAPTER II 



DIFFERENTIATION 



12. Increments. In order to understand the manner 
of variation of a function as the variable varies, it is essen- 
tial to know how great a change in value occurs in the 
function for a given change in value of the variable. 
Change in value is termed increment ; i.e. the increment of 
the function is the change in value of the function corre- 
sponding to a given change in value or increment of the 
variable. 

The problem now arises : To calculate the increment of a 

given function. 

Let f(x) be defined for 
all values of x from x to 

X -^h the value of the func- 
tion is f{x + //), hence the 
increment of the function 
fix) corresponding to an 
x" increment h in the vanable 



O 







X 

Fig. 8 



a?+/i 



X IS 



fix + h)- fix). 



We shall represent the increment of any variable by the 
letter A (read ''delta") prefixed to that variable, thus 

If Ax = h, then Af(^x) = f(x + h)-f(x). 
23 



24 DIFFERENTIATION 

Rule. To find the increment of a ftmction^ calculate the 
new value of the function by replacing x by x -\- k and sub- 
tract the old value of the function from the new value. 









EXERCISE 3 




1. 


Find A;r2. 




A;ir2 ={x ^- hy - x^ = 2 hx -\- h\ 


Ans. 


2. 


Find aQV 




\x)~x-\-/i x~ x{^x-^h) 


Ans, 


3. 


Prove AV^ = 


y/x 


. (Ex. 4, page 22.) 

^h^\/x 




4. 


Find A log x. 










Alog^=log( 


x-\- 


h) log.r=log( ^ ^=log(i+J. 


Ans, 


5. 


Find A sin x. 









A sin ;r = sin (;ir + ^) — sin :r = 2 cos (^ + i ^0 sin i /i, 

from Trigonometry. Ans, 

6. Find A^. A^ = ^+* - <?^ = ^^(^'^ - i). Ans, 

7. Find A cos 2x, A cos 2 ;r = — 2 sin (2 ;r + /i) sin 7^. ^^^j*. 

8. Find AvTT^. / . , , , /— — • ^^^^' 

13. The Increment Quotient. While the increment of 
a function as found in the preceding article is of impor- 
tance, still more essential in any investigation is the 7'ate 
of change of the function, that is, the change in the function 
per U7iit cha7ige in the variable. 

If we form the quotient 



DIFFERENTIATION 



25 



UO 



we obtain the average rate of change of the function while 
the variable changes from x \,o x ■\- h. 

For example, the '' law of falling bodies," 
given in Mechanics, asserts that the distance s 
traversed by such a body falling freely from rest 
in a vacuum varies as the square of the time /, 

that is, 

i'= 16. 1 fl, 

the constant 16.1 being determined experimen- 
tally when s is measured in feet and t in seconds. 

Therefore Ai- = 16.1 (/ + /if - 16.1 fl, 

As 



or 



A/ 



= 16.1(2^ + //), since A^ = ^. 



t=s 



For example, the average velocity throitghoict 

As 
the third second is given by setting in — , / = 2, 



Fig. 9 /^ = I, and is 80.5 feet per second. 



At' 



EXAMPLES 

1. From Physics we learn that for a perfect gas at constant tem- 
perature the product of the pressure p and volume v is constant, or 

c , , \i) c 
pv — 2l constant c. i.e. p —- \ show that —^ — ^ — 

2. Show from Ohm's law, viz. current strength C equals electro- 
motive force E divided by the resistance R^ that for constant R the 
change of current strength per unit change of electromotive force is 
constantly equal to i -^ R. 

14. Derivative of a Function. In the illustration taken 
from the law of falling bodies given in § 13, let us propose 
to ourselves to find the velocity at the cud of tzvo seconds. 
Making /= 2, we have 



A^ 
At 



= 64.4 + 16. I ky 



26 



DIFFERENTIATION 



which gives us the average velocity throughout any time 
h after two seconds of falling. Our notion of velocity 
shows us, however, that by the velocity at the end of two 
seconds we do not mean the average velocity during one 
second after that moment, or even during y-^-^- or -^^^^ 
of a second after that moment, but, in fact, we mean the 
limit of the average velocity when h diminishes toward 
zero ; that is, the velocity at the end of two seconds is 
64.4 feet per second. Thus, even the everyday notion 
of velocity involves mathematically the notion of a limit, 
or, in our notation, 

Telocity = Limit f^l 

Thus, after / seconds have elapsed, the velocity of a 
falling body is 32.2 t feet per second. 

Again, let it be required to find the slope of the tangent 
at any point P oi 2l plane curve whose equation is given 
in rectangular coordinates x and j^'. 




FiG. 10 



The tangent at P is constructed as follows (Fig. 10): 
Through P and any point P^ on the curve near P draw 
the secant AB, Let the point P^ move along the curve 



DIFFERENTIATION 2J 

toward P^ the secant AB meanwhile turning around P. 
Then when P^ coincides with P, the secant AB becomes 
the tangent TP. 

Now if P is (x, y) and P^ {x + A;ir, y + Ay), the slope of 
AB\^ 

^ a SP' Ay 

tan a = —— = — =^. 

/^5 A;ir 

As /" approaches /* as above described, Ax will ap- 
proach zero as a limit, while approaches the angle PTO 
or 7 ; hence, at the limit, 

tan 7 = Limit (-—] = slope of tangent at any point J". 

For example, the slope of the tangent at any point of 
the parabola J = ;r2 + 3 is 2;r. 

Law of Linear Expansion. \il^ is the length of a rod at o"^ Centi- 
grade, and / the length at f on the same scale, then experiment estab- 
lishes the law of expansion 

1 = Iq-V at + bt\ 

a and h being constants. The coefficient of linear expansion at any tem- 
perature / is the increase in length per unit change in temperature, i.e, 

coefficient of expansion — Limit ( — ) 

We easily find, then, that 

coefficient of expansion — a '\- 2 bty 

and /. a — the coefficient of expansion at o^. 

Specific Heat of a Substance. The specific heat of any substance is 
the quantity of heat necessary to raise a unit mass of the substance 
one degree in temperature. If Q is the measure of the quantity of heat 
in unit mass, and / the corresponding temperature, then by definition, 

specific heat = Limit ( — ^ ) 



28 DIFFERENTIATION 

These examples show that we obtain an important new 
function of the variable if we can find the limit of the 
Increment Quotient when the increment of the variable 
approaches zero. This function is called the derivative 
of the function. 

Definition. The deriyatiye of a function is the limit of the 
quotient of the increment of the function anil the increment of tlie 
yariable when the latter increment approaches the limit zero. 

The step of finding the limit of ~'^^ ^ when A.r ap- 

A.r 

proaches o is indicated by changing the A's to ordinary 

^'s, so that -^4^ = Limit ( '}'^^ ] , or also, if 
ax \ l\x /A3'=o 

Aa, = /., ^ = Limit r^^^^^^^^^l . 

The symbol -^ ^ ' is read, '' derivative of fix) with 
dx 

respect to .r." This, being a new function of .r, is often 
written f {x\ so that also, 

Thus in the illustrations given, 

velocity = — , 

i.e. velocity is the derivative of the space traversed in the 
time t with respect to the time. 

Slope of tangent = — , 
dx 



DIFFERENTIATION 29 

i.e. equals the derivative of the ordinate of the point with 
respect to its abscissa. 

Coefficient of linear expansion = — , 

dt 

or the derivative of the length with respect to the te77iperature. 



Specific heat = 



_dQ 



dt' 



that is, equals the derivative of the quantity of heat in unit i7iass with 
respect to the temperature. 

Many more illustrations of physical magnitudes might be given which 
take the form of a derivative. 

We call — the sign of differentiation, so that the pre- 
dx 

fixing of — to any function of x means that the following 
process is to be carried through : 

General Rule of Differentiation. V. Calculate the 
quotient of the incremeut of the function and the increment of the 
variable {i.e. the increment quotient). 

2^. Find the limit of this quotient when the increment of the 
variable approaches the limit zero.* 

It must be emphasized here that the characteristic thing 
in differentiation is finding the Hmit of a quotient. From 
the standpoint of the Differential Calculus a function is of 
no interest if the limit mentioned does not exist. Func- 
tions possessing derivatives are said to be dijfercjitiable, 
and it is of prime importance to show, for example, that 
the elementary functions of § 2 are differentiable. 

* The student must notice that the limit of the increment quotient cannot be 
found by Theorem III, § 8, since the limit of the denominator is zero. 



30 DIFFERENTIATION 

15. Differentiation of the Elementary Functions 
x^y sin a?, log^ x, 

{a) To prove —-x^^mx^'^, 
ax 

Now L{x^) = {x '\- hy — x"^ if l^x^h. 

But {x + hy = x'^ + mx'^-^h + • • • + ^^ 

the terms not written containing powers of h, 
.-. A(;r^^) = rnx'^'-Vi + ... + ^^ ; 

A;r 
where again the terms not written contain powers of h. 
Putting ^ = o, we find 

(1) — (;r^^) = ;/^'^~\ 
dx 

(b) To prove — sin ;r = cos x, 

dx 

Since A sin ;ir = sin (x + h) — sin x 

= 2 cos {x + I /i) sin ^ ^ (§ 12, Ex. 5), 
we find 

A sin ;r __ 2 cos (x + ^ fi) sin \ h 

Ax h 

= cos {x + \ k) . ^^^1 ' 

But Limit (5i^) =1 (§9), 

and Limit (cos {x + \ /i))h=.o = cos x 

(since cos.r is continuous, § 6), 
so that we may apply the theorem II, § 8, and we have 

(2) — - sin;ir= cos;tr. 
ax 



DIFFERENTIATION 3 ! 



(c) To prove — log^ x = log« e - 



dx 
From § 12, Ex. 4, we have 

A 



since the introduction of the exponent - is, by the princi- 

h 

pies of logarithms, equivalent to multiplying the logarithm 

itself by that exponent. 

/ Jt\- 
Now f I + - j^ is the expression of § ii if we write in that 



expression ^ = 7- 
h 

Also Limit 



= 00, 

/i=0 



hence Limit 



i+- 



m =Limitrfi+iYi =.. 

xj J;i=o LV zj j^=oo 

.-. Limit flog^ [\ + ^)^1 ^= log« e, 

(since log ;r is a continuous function), 
and we have 

(3) — log«;r=log«(?-. 

dx X 

Formula (3) becomes most simple when a^ e, for then 

d 1 I 

— log,^ = -. 
dx X 

Logarithms to the base e are called natiiral logariihuis or Napieria7i 
logarith77is (§ 10), and the factor loga^ in (3) is called the uiodtdiis of 
the system whose base is ^, i.e. the miinber by which natural logariihuis 
must be midtiplied in order to obtain logarithms to a7iy given base a. 

We write M — modulus = loga e. 



l^ 



DIFFERENTIATION 



For example, the modulus of the common system of base lo is log.Q^, 

and 

logio^ = 0.43429 
to five decimal places. 

If in (i), (2), and (3) we write tc for x, we have 

(4) -— - u*^^ = mu^^~^ : -^ sin t^ = cos 1/ : -^ loga ^ = loga ^ — • 
^^^ du du du u 





EXERCISE 


4 






1. Differentiate with respect to x. 








(a) X- -]- ^ ax -i- d. 






Ans. 


2r+ 3^. 


<*> .%■ 






Ans. 


a 


{x+by 


(0 ^^' (cf. Ex. 4, p. 22). 






Ans. 


I 



2. Prove — cos ;r = — sin ;r. 

dx 



3. Prove — vTT^ 



4. Prove ^f^y-iL. 

5. Prove — (0^) = C, if C is any constant 

6. From the law of falling bodies 

^•=16.1/2, 

ds 
we found (§ 14) ^ = 32.2 /, 

dt 

or velocity = 7^ = 32.2 1, 

Prove — = 32.2. 

dt ^ 

What does ^ = Limit f :^^ 

dt \ StJst=o 

represent in Mechanics ? Acceleration. Ans. 



DIFFERENTIATION 



33 



7. Find the velocity and acceleration of the motion defined by 

(i) s = at + I gt'^ \ Ans. V=a-\-gt\ accel. =^. 

(2) s = at — Igt'^'^ Ans. V=a—gt\ accel.= — ^. 

8. Find the slope of the tangent to y = 6 x— x'^ 2i\. the origin. 

Ans. 6. 

16. Certain General Rules. We prove in this section 
several important rules for differentiation of a general 
character. 

Let the variables u, v, w, etc., be functions of the vari- 
able X. 

I. To differentiate any algebraic stint of these variables. 

For example, to find — {ti -\- v — w), 
dx 

Now 

= A?/ + Lv — Aee/, 

A(t^ +v— w) _At^ Av Aw 

Ax Ax Ax Ax 



Since Limit 



'Au 
Ax 



AX=0 



4^, Limit r^l =^, 
dx L^;rjAx=o dx 



Limit 



' Azv 
A^ 



A2-=o dx 



we may apply I, § 8, and we have 
(5) 



d / ^ ^ ^__dtc dv dw 
dx dx dx dx 



n . To differ en tia te a p rodtict. 

For example, to get — {tiv\ 
dx 



EL. CALC. 



34 DIFFERENTIATION 

Now A{uv) = {t^ + A«)(z/ + Av) — uv, 

= uAv + vA2^ + An Avy 

A(uv) Av , All , . Av 
Ax A^ A;r Ax 

Since Limit r^l = ^, Limit f^l =$. 

Limit [A«]^^^ = o, 
we may apply I and II, § 8, and obtain 
/^\ ^ / \ dv , dii 

To find — {iivw), consider tivw as made up of the two 
dx 

factors tiv and w ; then, by (6), 

or by (6) again, 

/^x ^tt' , dv . du 

(7) = /^z/ --- + wii — - + wv — - 

^jur ^^ ^;r 



dx dx dx 



III. To differentiate a quotient. 

Since A T-^ = -^^ "^ ^^^ ^^ - "^^ "^^^ ~ ^^ "^^ 



V + Av V v^ + V Av 

All Av 

V tc ■ 

A fii\ __ Ax Ax 

Ax\v)'~ z^2 + vAv 

Then since Limit [-i^^ _^ ^^^j^^^^_= .,2^ ^^ j^^y apply I, 

II, III, § 8, and have 

die dv 
\P) d (ii\ dx dx 



dx \vj v^ 



DIFFERENTIATION 3 5 

From (6), (7), and (8) we have the rules : 

I. The deriyatiye of an algebraic sum of any number of yari- 
ables is equal to the same algebraic sum of the deriyatiyes of the 
yariables. 

n. The deriyatiye of a product of any number of yariables is 
equal to the sum of all the products formed by multiplying the 
deriyatiye of each yariable by all the remaining yariables. 

in. The deriyatiye of a quotient equals the denominator times 
the deriyatiye of the numerator minus the numerator times the 
deriyatiye of the denominator, all diyided by the square of the 
denominator. 

To these we may add the following : 

IT. The deriyatiye of a constant is zero. 

Y. The deriyatiye of a constant times a yariable equals the 
constant times the deriyatiye of the yariable. 

Rule V comes from (6) and IV, if we place ti equal to a 
constant. 

EXAMPLES 

1. Workout — (I +;r2)(i -2;r2). 

dx 

Rule II is first applied, and we get 

ax ax ax 

By Rule I, ^ ^i - 2x-^) = ^ (i) - -^(2x% 
ax ax ax 

Since by V, — (2 x'^) = 2 —x\ and from (4) § 15, —x'^ = 2x, 
dx dx dx 

we have finally, 

^ (l + :r2)(i-2x2) = (i+;r2).-4^+(l-2jr2) .2;r=-2^'(l4-4^'2). 



36 DIFFERENTIATION 

2. Workout Af^B^]. 

dx\\ogtxJ 

Rule III we use first and find 

^ , . loge^t'— - sin X - sin;ir — log^x 

a I s\nx\ _ ax dx 

dx \\og,x) ~ (loge^)2 



d . d 

— sin;f = cos;r, -— 
ix dx 

d /sin;r\ x zo^ x\oZe ^ — ^'^'si x 



By (4) § 1 5, ^sin ;r = cos x, — loge r = 1 . 



/ / sin ;r \ _ 
'x\\Qg^x] 



dx \ loge ^1 X (loge ^) ^ 

EXERCISE 6 

Prove the followino: differentiations : 



1. 



^x(i-x^=i-2x. 3. j^/siii^U£^os£-_sin^^ 



^:r ^ ^ dx\ X I 



5 <^ / ^"* '\ _ ;f"*-^(;;/ + x^ (m — n)) 
' dx\i +X'') ~ (i -hx'^y 

6. — (;ir"* log ;ir) = x"^~^(i + ;;/ log;r). 



d_ I 2X \ ^ 2(1 -hx^) 
dx\i-x^) (i-:r2)2' 



^(^-] = !i:,n-i, 9. ^/^^U- 



/ a 



dx\al a dxXx'^i x"^-^^ 

10. — ;t-"»(i -;r)" = a'"»-i(i -;r)"-i(;;/ - (;;/ + n)x). 
dx 

Special attention should be given to the following: 

11. Find — tan /^. Since tan // = , 

du cos u 

, d ^ d /sin/^\ 

we have — tan u — -—\ • 

du duxcosul 

Applying III (4), § 15, and Example 2, § 15, we find 

— tan^ = sec^^. 
du 



DIFFERENTIATION 37 



Prove 



12. — cotu = — cosec^/^. 
du 



13. — sec// = sec// tan ^/. (Put sec//= . ) 

du \ cosu J 

14. — CSC// = — CSC// cot//. 
du 

17. We come now to two most important rules. 
Differentiation of Inverse Functions. Suppose j is a 
function of x^ i,e, in symbols 

(9) y=A^\ 

Then it is usually possible inversely to calculate x when 
values are assumed for y, i.e. we may choose y for the 
independent variable instead of ;r, so that by solving (9) 
for X we obtain 

(10) ^ = </>(j). 

Then f{pc) and ^(^y) are called inverse functions. 

Example. \i y — a^, then x = loga/ ; that is, a' and loga/ are 
inverse functions. 

Let now A;r and Ay be corresponding increments of x 

and jK, so that Ax and Ay vanish together, since we are 

dealing here with continuous functions. Then the incre- 

. Ay Ax ,. . , 

ment quotient is -r^ or -r— , according sls x or y is taken 

for the independent variable. 

XT 1 , . ,. . Ay Ax 

Now by multiplication, — ^ • -r—= i, 

hence Limit (-r-^) . Limit (-r-^j =1, 



38 DIFFERENTIATION 

by II, § 8, since, as above emphasized, A7 and Ax vanish 
together. We have, therefore, in the derivative notation, 

dy dx 



^^./^=^'°'^^°'^^^' 



dy~ 


- 1 * 
cly 




dx 



(11) 



yi. If 2/ is a function of x^ and inversely x a fanclion of 2/, then 
the deriyatire of x with respect to y equals the reciprocal of the 
deriratiYe of y with respect to x. 

Differentiation of a Function of a Function. We have 
seen by (4) how to differentiate with respect to x the ele- 
mentary function sin^. Suppose we wish to find 

-f sin(i+^2)^ 
ax 

for which the rule (4) does not suffice. We then introduce 
the variable u = i + x'^, and setting j = sin(i +;ir2)= sin//, 
we have before us the relations 

(12) 7 = sin tc, u= I + x^, 

and we say j is a fimction of x through Uy i.e. y is 2l function 
of a function. 

Now, if Aj^, A//, and Ax are corresponding increments of 

y, ti, and x, then forming the increment quotients -^, — — , 
we have, by multiplication, 

(\%\ ^ A^^Ar 

^ ^^ An ' Ax Ax 

♦The student will not fail to notice that in (ii) the familiar property of a 

fraction, -= i h- - is suggested. But he must not forget that ~ is not a fraction, 
a ax 

Ay 
but merely the symbol for the limiting value of the fraction ^» 



DIFFERENTIATION 



39 



But the increments Aj/, Lie, and Lx vanish together, so 
that, by II, § 8, 

'^ • Limit r^) -^ ,, , 

\i^Uj^y^ \^Xj^^ \^Xj^^ 



Limit [-/-] • Limit (-^) = Limit ( ^ ) , 



or (14) 



dydy du 
dx ~ du ddo 



VII. If 2/ is a function of x through u^ then the derivative of y 
with respect to x equals the product of the derivatives of y with 
respect to u and of u with respect to x. 



'Thus in (12), since — sin/^ = cos^, 
du 



(i + :r2) = 2;f, we find 



dx 
^ sin (l + x'^) = cos /^ • 2 ;r = 2 ;r cos (i + ;ir2) . 

EXERCISE 6 

1. Show that the geometrical significance of (i i) is that the tangent 
makes complementary angles with XX' and YY' , 

2. If a material point /*, whose rectangular coordinates are x and y^ 
move in a plane, then x and / are functions of the time t. Now the 




horizontal component v^ (see Fig. 11) of the velocity v is the velocity 
along OX of the projection M of /*, and is therefore the time rate of 



40 DIFFERENTIATION 

dx 
change of x^ox v^— — . In the same manner, the vertical component 
, dt 

V.2 equals ^; and since 
we have 



'-im-m 



For the direction of the velocity, tan y = -^, or 

dy dx 
tan y = -^ -r- — - . 

dt dt • 

3. Prove that the equations x = a cost, y = a s'mt define uniform 
motion in the circle x'^ 4-/^ = a-. 

4. If the coordinates (^,y) of a point on a curve are functions of a 
variable 0, show that 

^'^^ dx~ dd ■ dd' 

(Use (14) and then (n).) 

d - 

5. To find — (;f') when q is any positive integer; t.e, to differ- 

dx 

entiate any root of x. 
1 
Put u — x^, then ;ir= ?/?; hence, by (4), § 15, 

and by (II) "^'^ - ^ 



^;ir ^/^«-i 



But «^~^ = ;r 5 = X *, 



dx q dx q 



i.e. />^^ sanie rule holds for roots and powers of the independent 
variable* 



DIFFERENTIATION 



41 



6. From (4), § 15, and VII, we have 



dx die dx dx 



(16) 



d ' d ^ ' s. die du 

— sin ^ = — (sin u) — = cos u — 
dx dti dx dx 



du 

dx 



d d ^du 



7. To find ^{x~')^ 
dx 



Letting u = x^ (Example 3) , we have 

p 
uP = x^, 

.-. ^ (x^ =^7iP= puv-^ ^ ( Example 6), 
dx dx dx 

= ^.^.^^(J), 

— fix ^ . - ;r? =^x'i . 
Hence the rule 0/(4), § 15, for powers holds for any commensurable 



exponent. 



du 
dx 



8. To prove -7- arc sin u — ■ 

"^ dx Vi - u^ 

Placing/ = arc sin ;^, we have inversely, 

u = sin/. 

.-. ^=005^, andby(i5), ^ = -L 



dy 



du cos J/ 



But 
Hence 


cos/ = Vi — sin-/ = Vi — ^/2 

dy _ I 




and by (16), 


du Vi - w2 

dy d . dx 

^ — -J-- arc sm u = — ^ . 



>/I - «3 



42 DIFFERENTIATION 

du 

rs r^ d ^ dX 

9. Prove -y- arc tan u — 



dx I -\- u^ 

(Remember sec^/ = i + tan^/.) 
du 



10. Prove -T- arc sec u = • 



d^ uy/u^ - I 

11. Prove ^a^ = a^ log, a — • 

Putting / = ^", we have inversely, 

u = loga/. 

... ^ = log..l (§15,(4)). 
dy y 

dy _ y _ d" _ 
du loga e \oga e 

dy d ,, , , <^^ ♦ 

.•. -^ = — ^« = ^« log, a 

dx dx dx 

In particular, — ^» = ^« — • 

Example 1 1 s-hows that the exponential function ^' possesses the 
remarkable property of being its own derivative, for 

^ r ^dx 

— e^ ^= e^ — = e^. 
dx dx 

In general, if ^ is any constant, then 

(i) ■ — ^«^ = ^^% since — {ax) = a ; 

dx dx 

that is, the derivative of the function e'^'^ is proportional to {i.e. a times) 
the function itself. For a reason now to be explained, the function e^^ 
is said to follow the Compoimd Interest Law. 

If P dollars be drawing compound interest at r per cent, then in the 

r 

time A/ the interest is /'A/, and hence the change in P or A/* is 

1 100 ' ^ 

given by 

LP^—P^t, or ^^-^P 
loo A/ loo 

♦ From the principle in logarithms, loge a = — - — 

loga^ 



DIFFERENTIATION 43 

Now suppose the interest to be added on continuously^ and not afier 
finite intervals of time A^, i.e. we make A/ approach the limit zero, and 
conceive of P as increasing continuously ; then 

dt 100 

so that a sum of money accumulating continuously at compound inter- 
est has precisely the property above enunciated in (i), viz. its derivative 
is proportional to the sum itself. 

18. From the examples in Exercises 5 and 6 and the 
Rule VII, we deduce the following fundamental formulae 
for differentiation : 

VIII. ^u^ = mum-i ^ (ni any commensurable number), 
doc cloc 

du 

dx ^^"^ *^ " ^^^"^ ^ IT* 
X. -—a^ = a^ logc a~{a any positive constant). 

CiX' (tx 

XI. -^ sin w = cos 1/ ^. 
dx doc 

XII. ^cosw = -sini/^. 
doc doc 

XIII. -^tant^ = sec2w^. 
doc dx 

XIV. -^ cot w = - csc2 u ^. 
dx dx 

XV. -^ sec t^ = sec w tan w — . 
dx dx 

XVI. -^ CSC w = - CSC w cot u — . 
dx dx 

du 

XVII. -^arcsinti:^ 1^^ . 
dx Vl - ii2 

du 
XVIII. -^arccosii- "^^ 



rfX Vl - ti2 



44 DIFFERENTIATION 



du 



XIX. -^arctanie= ^^ 



doo 1 + 1/2 

du 
XX. -^ arc cot w = - ^ 



cia; 1 + ti2 

XXI. -— arc sec u = 



doo u^iV^-1 

du 
XXII. 4^arccsct/ = -- ^ 



doo uVu2-l 

The formulae and rules I-XXII the student must memo- 
rize. With their aid differentiation of the commoner func- 
tions is made rapid and easy, but perfect famiharity with 
them is indispensable. 

To show the application of the rules three examples are 
now given : 



1. Find 



AfJL 



By III, ^( ^-^^ ^^ ^^ 



By I and IV, -^(i -x) = -i; 

dx 

from VIII, —(I +;r2)* = i(i +;r2)-i^(i j^ x% 

dx 2 dx 

and since — (i + jr^) = 2 ;r, we have 

dx 



d_ / I - X \ _ Vi + ;tr^. - I -(i - ;r)(i + x^) 



•2^~ir , 



X 



dx\vr^^y (1+^2) 

To simplify, multiply numerator and denominator by 

(i+;f2)i 



DIFFERENTIATION 45 

Then, since (i + x^)^ = i, we have, reducing, 
A. / I — -^ " N _ _ I + ^ 



2. Find ^ log^V^^^^. 

ax ^ I + cos ;ir 



For convenience, set / = loge \ — r 



— cos:if 



f cos;r 

Since logW l!|!cos^ "i ^^^ ^^ ~ ^^^ ^) " J ^^SC' + ^^^ ;ir), 
then by I and V, 

ify I d , . .id,,, . 

^=2 ^log,(i - cos^) --^log,(i + cosx). 

Applying IX, we have 

-— (i— cos;r) -— (i4-cos;r) 

±=l^ I ^ , and by XII, 

dx 2 I — cos X 2 1+ cos X 

dy I f sin X sin ;r \ sin :r _ i 



[ / sin :r sin ;r \ _ 

i\i — cos X I + cos x) ~ I 



dx 2 V I — cos X I + cos xj I — cos^ x sin x 



. ^ 1^^ -* I - cos or 



cos X sin :r 



d fe^ - ^• 



3. Find ^ arc tan 



(e^). 



Setting the function equal to/, we have, by XIX, 

dy _ dO\ 2 I _ 2 



2 



:)' 



dO fe^-e-^Y 4-{-e^^-2-he 



(byX) 



2 

V 4- <?-^ 



46 DIFFERENTIATION 

EXERCISE 7 
Prove the following differentiations : 

1. — {x- -f i) va'^ — X — '- . 

dx 2(;t-3 - x)h 

2. ^V ^ ^ ^' = ^ . 

' dx^i -X (I _ ;t^) y./Y^r^i 

3. -^f ^ W ? . 

4. ^ / 3-^^ + 2 \ 2 
^-^'L-(^-3+i)l/ x\x^-\-i)i 

5. — (I - 2;f + 3;t-2 - 4:r3)(i + 2')'^= - 20;r3(i + ;f). 

6. —(I - 3;r2 + 6;r^)(i + :t-2)3 = 60 x'\\ -V x'^f, 
dx 

7. ^(5-'2x)^ 2Cr+ I) S"'^'' log, 5. 

8. — Ji'"^' = x'^-^a'^in + jr log, ^) . 
dx 

9. / r^og«-^+ iog.(i - x)i = J^^. 

dxV- \ — X J (i— :*•)* 

10. ±Ax^-^-^ + ^-^-^\ = ax^^. 
dx \ a aP- a^l 

11. <loge(^- + ^-)-^'~'"' 



dx e"" + ^~' 



12. ^(VJ-log,(l +VI')) = 

13. -^ tan2 5 (9 = 10 tan 5 (9 sec^ 5 (9. 
du 

14. ^ sin3 6cos0 = sin2 6> (3 cos2 ^ - sin^ 6). 
do 

^^' 4 log sec ^ = tan 6. 
do 



DIFFERENTIATION 47 



16. 4^(taii2 6-logsec2(9)=2taii3i9. 

17. i- sin ^/^ sin^ 6 = n sin^-^^ sin {n -f- i) 6, 
dd 

d x 

18. — arc sin (3 ;ir — 4 jr^) = ^ - . 

^■^ Vi — x^ 

■ye% d X a 

19. — arc sec - — 



dx a xVx^ - a^ 

r,r\ d I 2 

20. — arc esc — — 

dx 2 x~ — I Vi — 

ni d . I — x'^ 2 

21. — arc sm 



dx 1 + x'^ I + ^^ 

22. -^arctan.-*'+'^- ' 



dx I — ax I -{• x^ 

23. — arc cos - — 



dx e"^ + ^-* e^ -\- e- 



24. — arc sec -v' — - — = 

dx ^ I + ^ 2 V I — . 



25. 



d_ 
dx 



arc cot - + loge \ = — -. 



19. Differentiation of Implicit Functions. If an analytic 
relation is given between two variables not solved for either 
variable in terms of the other, then either variable is said 
to be an uriplicit function of the other. 

For example, in x^ — )^ + 9 = either x or y is an im- 
plicit function of the other variable. 

In such a case either variable may be chosen for the 
independent variable, and if we can solve explicitly for the 
other (as in the above example for y, giving -j' = ^x^ — 9), 
then we can differentiate as before. But it is generally 
better not to solve the equation, but to differentiate the 
given relation as it stands. 



48 DIFFERENTIATION 



Thus, to find -~ from 
dx 



^ xy -\- 2j2 



^^^"^ l-^^^>-3£(^-^>^^l-(-^^>=£3. 



"^-3(->' + -^|-) + 4;'S=o. 



and 



cfy ^ 2x- ^y 
dx ^x-4y' 



To justify this process is beyond the hmits of this text- 
book. One thing is to be noted, namely, that only those 
values of the variables which satisfy the original relation 
can be substituted in the derivative. 





EXERCISE 8 






dy 
Find -f- from the foUowinor 
dx * 


equations : 






1. y'^ -2xy = a^ 






Ans. 


dy _ y 

dx y — X 


a^ W- 






Ans. 


dy b'^x 
dx~ ay 


3. ax- -^ 2bxy -^ cy- -\- 2 


fx+ 2gy + h 


= 0. 

Ans. 


dy ax -\- by + f 
dx~ bx -\- cy -\- g 


4. x^ + y^ - ^axy = o. 






Ans, 


dy x^ — ay 
dx~ y^ — ax 


5. x^ -\-yi = ai. 






Ans. 


dy y^ 
dx- ^\ 



6. Given r = ^(i — cos 6) ; show — = ^ sin 0. 

dv 

7. Given r2 = ^2 cos 2 (9 ; show ^ ^^ - ^' s^^ 2 g 

dQ r 



DIFFERENTIATION 49 

20. Derivatives of Higher Orders. Since the derivative 
of a function of a variable x with respect to x is also in 
general a function of x, we may differentiate the derivative 
itself, that is, carry out the operation, 

dx\dx 

This double operation is indicated by the more compact 
notation, 

and this new function is called the second derivative. In 
the same way, 

is the third derivative, and in general, 

is the «th derivative of f{x), that is, the result of differen- 
tiating f{x) n times. The following notation is also used, 

The operation of finding the successive derivatives is 
called successive differentiation. 



EXERCISE 9 
1. Given f{x) = 3 ;r^ - 4 jr^ -f 6 ;r — i, 

then f(x) = 12 r^ - 8 ;t- 4- 6 ; 

/"(x)= 36 x^ - 8, etc. 

EL. CALC. — 4 



50 



DIFFERENTIATION 

2. Given /(r) = ^^ : prove /^'*^(;i') = ^"^'. 

(- iY \n - I 



(I -xy 



3. Given f{x) = loge ( i - ^') = Pi"o^'e /^») (^) = ■ 

4. Given / = jr^ loge ^'; prove -j— - -• 

^3i/ 2 COS ^' 

5. Given v = loge sin ^- ; ^nd — - = ^— — 

6. Given / = e^^ C-^'- - 3 -^ -^ 3) • ^^^ ^3 = 8 ;r^^2x. 

7. Given - -|-tt. = i. or b-x- + ^y- = ^-^-, to find -/-.- 

a' u~ ctx 

From Ex. 2, Exercise 8, 

dy _ b-x ^ d-y _ ^ dx^ ^ dx^ ^^ 

^;r " ^-y' ' * dx'^ ~ a^y^ 

, , ., dy 

a-b-y-b-a-x^^ 

or Tr^= r"o ; 

then subslitutino: for -^ and reducing:, 

d'-y _ b'\dY^b\x'-) _ ^ 

8. From y^ = 4 Ar. find ^; = - lA' = - -• 

9. From y — 2 xy = a-^ prove — 



^;r"^ (y — x)^ 



CHAPTER III 
APPLICATIONS 

21. Tangent and Normal. For all applications of the 
Calculus to Geometry the fact established in § 14 is of 
fundamental importance, viz. 

Theorem. The value of the derivative of y zvith respect 
to X foii7id from the i^qication of a citrve in rectangular coor- 
dinates gives the slope of the tangent at any point on that 
cnrvCy or 

-2. = slope of tangent. 
dx 

If we wish the slope at any particular point (-^^y), we 
have to substitute x^ and y^ respectively for x and y in the 

general expression for --^- Let (-7-) be the value of 



, dx \dx) 

-^ after this substitution, then we have from Analytic 
dx 

Geometry, 

Equation of the tangent at {x\ y') is 

(17) ^-^' = (M)'^'^-'''^- 

Since the normal is perpendicular to the tangent, and 
from (I I), (±^=' we find 



Equation of the iiormal at (x\ y') is 

(18) _y_y = _(^^y(^_;,') 

5» 



52 



APPLICATIONS 



tiating, -J- — - 



EXERCISE 10 

1. Find equations of tangent and normal to the parabola j2= 4 ;f+i 

at the point whose ordinate is 3. 

Substituting 3 for/, we find x — 2^ hence (^,y) is (2, 3). DifFeren- 

~ ~" '** \dx] ~3* 
Ans. tangent, 2;r— 3/4-5 =0; normal, 3 ;ir + 2/ — 12 = o. 

2. Find equation of tangent to the ellipse U^x'^ + aP'y'^ = aW at 

{x\ /O- ^^^' ^^^'^ + ^Vy = ^'^^'^^ 

3. Show (Fig. 12) that the subtangent M^T^ = -/'(—) , and the 

/dv\' ^^y' 

subnormal M^N^ = /M -^ J . 

4. Prove that the subnormal in the parabola y*^ — ^px has the con- 
stant length 2 p. 

22. Sign of the Derivative. An important question is 
the following : 

Is the function ina'casiitg or decreasing as the variable 
passes through a given value a ? 

The phrase " passing through a '' is understood to mean 
that the series of values assumed by the variable is an 




M, T, 



Fig. 12 



increasing sequence including a, i.e, on the graph of the 
function we proceed from left to right. In Fig. 12, as we 



APPLICATIONS 53 

pass through P^ the ordinates are decreasing, while at P^ 
the ordinates are increasing, and since the ordinates repre- 
sent the values of the function and -^ ot/'(x) is the slope 

ax 

of the tangent, we have the result : 

The function f{x) is increasing or decreasing as x passes 
through a according as f{a) is greater or less than zero. 

At P^ and P^ (Fig. 12) the tangent is parallel to XX' ^ and therefore 
f'{x) vanishes at these points. For such values of x, therefore, the 
rule just given does not enable us to answer the question proposed. 

If, now, for any value of Xy say ;r=^, the second deriva- 
tive — ^, or f'{x\ is positive, then as x passes through a, 
dx^ 

the first derivative fix), or tan 7, must be an increasing 
function of x, i.e. 7 must be increasing as x passes through 
a ; and therefore as we pass along the curve from left to 
right, the tangent is rotating coimter-clockwise, and the 
curve is accordingly concave upward {diS at {a)y Fig. 13). 





(&) 



Fig. 13 



On the contrary, if f^\a) < o, the reasoning shows the 
tangent to be rotating clockivise as we pass along the graph 
through X = a, and hence the curve is concave downward 
(id). Fig. 13). 



54 APPLICATIONS 

The result is : 

A cuTue is concave tcpwai'd or doivnwai^d as x passes 

through a according as the value of the second derivative 

d'^y 

-7^ for X =^ a ts greater or less than zero, 

d^y 
As before, if -f^ = o, the rule just given does not enable 

us to decide. If -f^ = o for x= a and changes sign as x 

passes through ^, then at ;r = ^ we have a point of inflec- 
tion {P^ and P^y Fig. 12). 

EXERCISE 11 

1. Show that the following functions are either ahvays increasing or 
always decreasing, and draw the graphs in each case : 

(^) tan;i'; (d) e"" \ {c) \ogx\ (d) -• 

2. Show that / = sin;i' has a point of inflection at each intersection 
with^^'. 

3. Determine the points of inflection of / =(x — ay + d. 

Ans. (^, b). 

23. Maxima and Minima. A function /(;r) is said to be 
a maximnm for ;r = ^ when f{a) is the greatest value of 
/(jr) as X passes through a, 

A function /(;r) is said to be a minimum for ;r = ^ when 
> f(a) is the least value of f{pc) as x passes through a. 

In other words, a maximum value is greater than any 
other in the immediate vicinity, and similarly for a mi7ii' 
mum value. It is not to be inferred that a maximum value 
is the greatest of all values of the function ; on the con- 
trary, a function may have several maxima. 



APPLICATIONS 



SS 



Graphically, at a maximum we have a higliest point 
{P^ and /^3, Fig. 14), at a minimum a lowest point 
(7^2 and Z^^). 




Fig. 14 



Since, by definition, if f{ci) is a maximum, f{x) must be 
an tncreasuig function for x<a and a decreasing function 
for x>a, we have (§ 22) : 

Theorem. If f{a) is a maximum value of f{x\ then 
the first derivative f'{x) must change sign from positive to 
negative as x passes through a. 

By similar reasoning for a minivtum^ we find a change 
in sign fro7n negative to positive must occur in fix). 

In either case, therefore, /'(^) must change sign. If we 
now assume that fix) is continitous for x = a, we see that 
f{a) = o ; that is, the tangent at a highest or lowest point 
must be horizontal {P^ and P^ in Fig. 14). If, on the con- 
trary, f\x) is not continuous for x = a, then the change in 
sign occurs by passage through 00 ; i.e. the tangent becomes 
parallel to YY' , as at P^ and P^. This case is, however, 
of minor importance, and is omitted from further con- 
sideration. 

Furthermore, if f\a)< o, the curve at ;r = <? is concave 
downward, and we have a highest point (/^j), while 
f\d)>o indicates a lowest point {P^)- 



56 APPLICATIONS 

We have therefore the following 

Rule for determination of Maximum and Minimum 
values of a function f{x). 

Find the first derivative f{x), and get the roots of the 
equation f\x) = 0. 

First Test. If f (x) changes sign as x passes through 
any root a of the equation f(x) = 0, then f(a) is a maxi- 
mum or minimum value according as the change is from + 
to — , or from — to +. 

Second Test. Find the second derivative f (jr) ; then, 
if a is any root of ^'(jr) = 0, f{a) is a maximum if f"(a) < 0, 
and a minimum if f"(a)>0. If, however, /"(a) = 0, we 
must use the first test. 

EXAMPLES 

1. Examine the function x^ — ;^x^ — gx-\-^ for maxima and minima. 
Placing /(x) =x^ — ^ x'^ — g X -{- Sy 

then /' (x)=;^x^— 6x—g, 

and the roots of ^x^^— 6x— g = o are x= ^ and — i. 

Now f'(x) = 6x-6, and /''(3)=I2, /X-i) = -l2, 
hence by the Rule, Second Test, 

y(3) = — 22 is a minimum value, 
and /"( — i) = ID is a 77iaxi7nu7n value of the function. 

The student should draw the graph. 

(x — I V 

2. Examine the function -^^ ^ for maxima and mmima. 

(^ + I )« 
Here f(x^^ ^^~ ^^^ 

Differentiating and reducing, we find 



APPLICATIONS 57 

The roots of/'(:f) = o are therefore x= \^x—^. We now apply the 
First Test, since it is unwise to form the second derivative. 
Taking account of the signs only, we have 



When*;r<i,/(;r) = - ^ )( ^ =- 
When x> i, f\x) = - ^ + )^~^ = + 

When x< 5, f\x) = - C + )(~) = + 
When x> 5, f(x) = - ( + H + ) = _ 



Hence f{x) is a 
minimum when x—\. 



Hence /' {pc) is a 
maxi7Jium when x= ^, 



.-. y( 1)1=0 is a minimum, and /(5)=:^ a maximum value of the 
function. 

EXERCISE 12 

1. Examine the following functions for maxima and minima : 

(a) x^— ^x+ ^. Minimum value 1 1 . Ans. 

(d) Max. value ^, min. value — 1. Ans. 

^ ^ I -{-x^ 

{c) 6x+ ^x'^— 4x^. Max. value 5, min. value — |. Ans. 

(^) x^— 2 ^^-\- 6 or. No max. or min. values. Ans. 

(e) ax'^ -^ 2dx-\-c, If ^ > o, min. value ^^-^ — , if ^ < o, 

a 

then is a maximum. Ans. 

a 



(/) 10 V8 X — x^. Max. value 40. Ans. 

This function is a maximum or minimum according as S x — x'^ is 
a maximum or minimum, hence f a constatit factor or a radical sign 
may be dropped in investigations of this sort. 

* We consider values of x differing only very slightly from the number on 
the right of the inequality sign. 

t If « is any j^olynomial in x containinfr fjo multiple factors, we may show that 
•yju is a maximum or minimum only when w is a maximum or minimum. For if 



58 APPLICATIONS 

2. Divide the number a into two such parts that their product shall 
be a maximum. 

Hint. If x is one part, then ^ — ;r is the other, and the function to 
be examined is x(a — x) or ax — x^. Equal parts. Ans. 

3. Divide the number a into two such parts that the product of the 
;^/th power of one and the nth power of the other shall be a maximum. 

In the ratio m : n. Ans. 



24. The subject of Maxima and Minima is one of the 
most important in the applications of the Calculus to Ge- 
ometry, Mechanics, etc. It is often necessary to derive 
the expression for the function to be investigated, and in 
testing this, attention should be paid to the remark in 
Example i (/) of the preceding exercise. 

EXERCISE 13 

1. A box with a square base and open top is to be constructed to 
contam io8 cubic inches. What must be its dimensions to require the 
least material ?* Base 6 inches square, height 3 inches. Atis. 

2. The strength of a rectangular beam varies as the product of the 
breadth d and the square of the depth ^. What are the dimensions of 
the strongest beam that can be cut from a log whose cross section is a 
circle a inches in diameter ? f Breadth is ^ ^ v^J inches. Ans. 



/{x) = V^, /' W = -^ ^. and /- W =- -1^ ^ + -1^ ^. so that /\x) 

vanishes only if — = o, and then/"(j»r) has the same sign as . 

dx dx'^ 

* Hint. Let x be the side of the base, y the height, then x'^y — 108, i.e. y = 

x'^ 

and since the material is x^-\-4 xy, we find by substituting for y the function 

X 

t Hint. The strength therefore equals hd"^ multiplied by some constant, which 
maybe dropped by the remark of § 24. But d'^ = a^ — b'^\ hence the function is 
b{a'^—b'^), b being the variable. 



APPLICATIONS 



59 



3. Find the dimensions of the stiffest beam that can be cut from 
the same log as in 2, given that the stiffness varies as the product of 
the breadth and the cube of the depth. Breadth J a inches. Ans, 




Fig. 15 
4. The equation of the path of a projectile (see Fig. 16) is 



y — X tan a 



gx' 



2 7v cos^ a 



where a is the angle of elevation and v^ the initial velocity. Find the 



greatest height. 
T 



2^ 




Fig. 16 

5. Find the dimensions of the rectangle of greatest area that can be 
inscribed in the ellipse b'^x^^-a^y'^— aP-b^. Ans. Sides are ay/i and b^i. 

6. Find the altitude of the right cylinder of greatest volume inscribed 
in a sphere of radius r. ^l^j^^^^ ^ 2^, ^^^ 

V3 




6o APPLICATIONS 

7. Assuming that the brightness of the illumination of a surface 
varies directly as the sine of the angle under which the light strikes the 
surface and inversely as the 
square of the distance from the 
source of light, find the height 
of a light placed directly over 
the center of a circle of radius 
a when the illumination of the 
circumference is greatest. 

From Fig. 17, the bright- 
ness at P is given by 

K sin ^ _ KX _ X _ f x^ y. 

X^ 

Hence the brightness is a maximum when is a maximum. 

x—-^. Ans. 
25. Expansion of Functions. By actual division 

(19) -^— =l+X+X^+"'+X- + f-i— V+S 

^ I — ;r \i — xj 

where 71 is some positive integer. In this simple way we 

may find for the function an equivalent polynomial 

all of wJiose coefficients save that of x^'^^ are constants. By 
transposition (19) becomes 



(20) (l +;r + ;r2 + ;r3+ ... +;ir'*) = 

^ ^ I — ^ ^ ^ I — . 



-;r'*+\ 



Now let X be some number mmierically less than i, say 

;r=.5, and suppose we wish the value of correct 

within one one-hundredth, i,e. correct to two decimal places. 
Let us then determine for what values of n the term 

;j;n+i ^h^n X =^ X is less than .01, i.e. solve the in- 

I —X -^ 



equality — — — .5''+^<.oi. We find n>6. 



APPLICATIONS 6l 

Furthermore, if x is numerically less than .5, 

and x'''^^ are less than for ;r = .5, so that taking n = j 
{i.e. >6), ;r^<.oi for every value of ;r not numeri- 
cally greater than .5. And we now see from (20) that 
the function __ may be replaced by the polynomial 

\ -^ X '\- x^ + x^ + x^ + x" -\' x^ + x^' for all values of x 
numerically equal to or less than .5 if results correct only 
to hundredths' place are desired. 

Precisely the same reasoning holds for a7ty value of x 
numerically less than unity, since for any such value x'^^^ 
can be made as small as we please by taking n sufficiently 
great. But this reasoning does not hold for any value 
equal to or exceeding i numerically. We may then state 
this theorem : 

For any value of x numerically less than unity ^ the func- 
tion may be represented with any desired degree of 

accuracy by a stcfficiently great number of terms of the 
polynomial ^ 

I +x + x^ + x^+ •••. 

The Differential Calculus enables us to obtain a similar 
theorem for many other functions, as will now be explained. 
In all practical computations results correct to a certain 
number of decimal places are sought, and since the process 
in question replaces a function perhaps difficult to calcu- 
late by a polynomial with constant coefficients, it is there- 
fore of great practical importance in simplifying such 
computations. 



62 



APPLICATIONS 



26. Theorem of the Mean. If f{x) and f{x) are con- 
tinuous as X varies from a to ^, then there is at least one 
value of X, say x^, between a and ^, such that 



(21) 



fib)-fia) .,,. 
b-a -J ^^1^- 




Fig. i8 



In Fig. II, f{b)-f{d)^CB, d - a = AC, 

.*. -^ — -^ ^ ^ = slope oi ABy and at each of the points 

P^ and P^ the tangent is parallel to AB, and hence (21) is 
true if x^ is the abscissa of P^ or P^* 
Multiplying (21) out gives 



(22) 



f{b)=f{a) + {b-a)f'{x,), 



where it must be remembered a>x^> b. 

A more general theorem than (21) is enunciated as 
follows : 

If fix) and the {n + i) successive derivatives f\x\ 
f\x\ •••, f^'^'^'^^ix) are continuous when x varies from a 

* This proof of the theorem of the mean is not mathematically rigorous, but 
merely illuminates the significance of (21). The student should draw other figures, 
and especially such that the necessary conditions of the continuity of f{x) and 
f{x) fail. 



APPLICATIONS 63 

to by then there is at least one value of ;r, say x^, between 
a and b such that 

(23) Ab) ^f{a) + ^^^f{a) + ^^^f\a) 

\n + I -^ ^ ^^ 

The proof of (23) is beyond the scope of this book.* 
The student should, however, carefully note the law by 
which the expression on the right is constructed. 

Putting for b in (23) the variable ;r, we get Taylor's 
Theorem^ 

(24) f{x) =f{a) + ^^^f'{a) + ^^^f"{a) + •.• 

+ —, — -^ — /^'"'^^X-^i), where a<x. <x. 
\n + I -^ ^ A/^ 1 

Finally, setting <^ = o in (24), we find Maclainnn's 
Theorem^ 

(25) /(^)=/(o)+f /'(o)+^/"(o)+ - +g/'^'(o) 

+ , /^''■^^^(^i) where, o<x.<x, 

\n + I 

If in (23) we put b — a -\- x^ we obtain another form of Taylor's 
theorem, 

f{a + X) =f{a) + ^/'(^) + ^f'(a) + ... etc. 

This formula (25) gives/(A') in the form of a polynomial 
in X with constant coefficients save that of x"^^^, which, 
since x^ lies between o and x, is a function of x] that is, 

* An excellent discussion is given in Gibson's An Elementary Treatise on the 
Calculus, London, 1901, p. 390. 



64 APPLICATIONS 

we have the generalization of the example of § 25 as 

follows : 

A function f{x) for certain values of the variable * may 
be represented with a7iy desired degree of accuracy by the 
polynomial^ 

/(o)+^/'(o) +|/"(o) +g/"(o)+ - +g/<«>(o). 

By " expansion of a function " is meant the forming of 
this polynomial. Of course n is indefinite, and must be 
taken great enough to give the desired degree of accuracy. 
It is of greatest theoretical importance to determine for 
what values of x the polynomial represents the function 
when n is taken indefinitely great. This consists in exam- 
ining for what values of x 

""'K(i5T-^""<">).-=°' 

for this term is the difference between the function and the 
polynomial. 

EXERCISE 14 

1. Expand sin;r. 

Since f{pc) = sin x, and for x= o, f(6) = sin o = o ; 

then f{^^ = cos x^ and for x= o, f'(p)= i ; 

f'{x) = - sin r, /"(o) = o ; 

f"(x) = - cos X, f"{o) = - I ; 

f'^ix) = sin X, f^^Co) = o ; 

etc. etc. 

x^ x^ x' x^ ^ 
Hence sin ;r = ;r - , 1- , , h , etc. 

\2l \S_ [L \SL 

2. Show that the expansion of cos x is 

x'^ x^ x^ x^ 

cos ;r = I - , 1- , i-z- + ,-5- - etc. 

[2 [4 lA [8_ 

* Namely, for all values of x such that the " remainder" j — 37- /^**+^) C-*"!) is 
less than the limit of error. This question is often difficult to settle. 



APPLICATIONS 65 

3. Expand ^. 

Since f^x) = <?*, and all its derivatives are likewise e'^y while d° =■• i, 
we obtain ^, ^3 ^, ^.5 

Putting X = I, we find 

the expression given in § lo. 

The expansions of sin x, cos x^ and e"" are remarkable in that they 
hold for every value of x, positive and negative. 

4. Prove the following expansions : 

, ^ , , , , fx x^ x^ x^ x^ \ 

(a) ioga(i +^)=ioga^^Y"T + y"7"^y"**T 

,,x X X mini — \) ^ 7?i(7n — i)(7Ji — 2) „ 

[2 [3 

(<^) is the binomial fori Jiiila. These expansions hold onXy for values 
of X nu?nerically less than i . 

Taylor s Theorem (24) differs from (25) in that we are 
to consider values of the variable x near some given num- 
ber a^ since (24) is a polynomial in {x — a) in the same 
sense that (25) is a polynomial in x. It is evident that no 
greater difficulty arises in the application of (24) to a given 
function than has been already pointed out. 

27. Differentials. From (23) we are able to find an ex- 
pansion for the increment of a function in powers of the 
increment of the variable as follows : 

Write ^ = ;r + A.r, a=x, .*. d — a = AXy and (23) be- 
comes, after transposing /(x\ 

(26) fix + A^) -fix) = Axf'ix) + ^-^f'ix) +..., 

or (27) Afix) = fix) Ax A- fix) ^-^ + •••• 

Now, if we suppose A.r to diminish toward zero, the first 
term f{x)^x of the right-hand member will ultimately 

EL. CALC. — 5 



66 APPLICATIONS 

greatly exceed the sum of the remaining terms, since these 
contain higher powers of A;r. For this reason f{x)^x is 
called the principal part of the incremeitt of fix). Also, 
when we wish to emphasize the fact that the variable A;r 
is to approach zero as a limit, we write dx^ called differen- 
tial X, instead of A;r, and the principal part of the incre- 
ment /'(;r)^^ we call the differential of the function; that is, 
(28) df{x)^f{x)dx. 

The following definitions are fundamental: 

A differential (or infinite signal) is a vai'iable whose limit 
is zero. 

The differential of the independent variable is an incre- 
me7it of that variable whose limit is zero. 

The differential of the depertdent vaiiable is the princi- 
pal part of the increment of that variable^ and equals the 
product of the derivative and the differefitial of the inde- 
pende7tt variable (28). 

From (28), we see that if j is a function of x^ then 

EXERCISE 15 

1. Prove by (28) and (29) the following differentials : 

{a) d{ix'^)^(>xdx. (^) dW^x = ^ 

, 2V1 — X 

(b) d\og,x = —- (J) dsm2x=icos2xdx, 

{c) de'^e'dx, stc^(-\ 

(d) dx-^ = mx-^-^dx, (^) ^ ^^^ (3 " IF^ '^^' 

{h) U y = xlogeX, then dy =(i + loge x) dx, 

2. If y = uvy then 

dy=(u'^+v^)dx = 'u^dx-^v^dx, or dy = u dv -\- v du. 
\ dx dxl dx dx 



3. Show that 



APPLICATIONS 
V du — udv 



67 



'Ki)- 



V' 



4. State the rules I-V for differentiation in terms of differentials 
instead of derivatives. 

28. We may write {2J^ after replacing A;ir by dx, 
(30) A/(^) =f{xYx + dx-^i^ + /^ dx +...). 

Now, since by (28) f{x)dx is the differential of the 
function, (30) shows that A/(jr) and df{x^ differ by a term 
containing the factor dx'^. Such a quantity is called a 
differential of the second order ; in general, any quantity 
containing as a factor the product of two differentials is 
thus designated. 

The increment of a function differs from its differential 
by a differential of the second order, 

EXAMPLES 

1. Differential of a product uv. 



-§ 



udv 



D 



E' 



du 



B B' 

Fig. 19 

Let u = AB. V = ACy then 7iv = area A BCD. If du = BB\ 
dv — CC ^ then 

A(/^7/) = area AB'CD' - area ABCD 

= area CDC E -^ area BB'DE' 4- area DE'ED' 
— tidv -{- V du + du • dv. 
Now du ' dv is a differential of the second order, .*. principal part 
oi ^{tiv) is udv -\- vdu\ i.e- d{uv)~ udv + v du. (Cf. Ex. 2, § 27.) 



68 APPLICATIONS 

2 . Differential of a7t area . 




^ 



dx 



X' 



21 N 



Fig. 20 



Consider the area aAPM bounded by any cun'e. the axis XX' and 
the ordinates aA, MP. and call this area u. Then if MiV=dx^ Aw 
= area ^/^gA^- area ^^/^J/= 2.x^2.MPQN'. r. ^u=.ydx+ area/'Jg. 
But area PSQ <dx' dy. . • . PSQ - k dxdy^ where k is some number < i . 
Hence area PSQ is a differential of the second order, and .-. du-y dx. 

The differential of the area bounded by a?iy curve, the axis XX', and 
two or dictates is the product of the ordinate of the curve and the differ- 
ential of the abscissa. 

3. Differential of the volume of a solid of revolution. 

Let the solid be generated 
by revolving a curve APQ 
around XX', and denote the 
volume APA'F by v. If 
dx — MN., then At/ = volume 
AQA'Q - volume APA'F, 
or At/ = volume of the cylin- 
der PSP'S' -\- volume gener- 
ated by the curvilinear A PSQ. 
Now the volume of the cylin- 
der PSP'S' = iry-'dx, since y 
= PM = radius of base and 
rfr=r altitude. The volume 
generated by the curvilinear 
A PSQ < volume generated 
by the rectangle PRSQ, and 
this last volume = ttA^'^ • MN - ttMP^ • MN:=: 7r(2ydy + dy^)dx. 



f 


1 


B 


t 

dx 


Q 

\ 




s~ 


a I ; 




21 


N 


X 


V 


^ 


F 


I 







R' 



Fig. 21 



APPLICATIONS 



69 



We see therefore that At/ = tt/^ dx-\-2i differential of the second 
order, i.e. dv — iry'^dx. 

The differential of the volume of a solid of revolution generated by 
revolving any airve around the axis XX' equals tt ti7nes the product of 
the square of the ordinate and the differential of the abscissa. 



Oir^dVyQ^dQ) 




PCr,(?) 



Fig. 22 



4. Show that the differential of the area u bounded by a curve AP 
and two radii vectores OA and OP is given by du = \ r'^dd, where 
(r, 6) are the polar coordinates of P* 



CHAPTER IV 

INTEGRATION 

29. Indefinite Integral. Integration consists in finding 
a function of which a given differential expression, such as 

dti 
X dXy sin X dxy — , etc., is the differential. The function 
// 

thus found is called the integral of the given differential 
expression, and the operation is indicated by prefixing the 

integral sign j . Thus, since 

d{\x'^^^ X dx, .', ixdx = ^x^; 

j dx = X, I sin xdx= — cos x^ etc. 

In general, 

]f{x)dx 



fj 



means to find a function F(x) such that 
dF{x)=/{x)dx, 

i.e, f{x) = — F{x\ 

Constant of Integration. Since d{^\ x'^ + C) also equals 
xdx, no matter what the constant C is, we have 



/ 



X dx =^\x'^ ■\- C, 

where C is any constant whatever, called the constant of 
integration. We see, therefore, that a given differential 

70 



INTEGRATION 71 

expression may have infinitely many integrals, found by 
giving to the constant of integration different values. 
Thus 

^f{x)dx = F{x)+C, 

and since C is unknown and indefinite^ ^{^) + C is called 
the indefinite integral of f{x) dx. 

Of course, the same differential expression has an in- 
definite number of distinct integrals, but what has just 
been said shows that the difference of any two of these 
must be a constant. 

30. Rules for Integration. From Rule V in differentia- 
tion, if V is any function of x, and k a constant, then 

— (icv) = K — , i,e, d{icv) = k dv. 
dx dx 

Integrating, we have, since if two differential expressions 
are equal so are their integrals equal, 

I icdv = \ d{Kv\ 

or, since j d{icv) = tcv, 

fcv = I fc dv. 

But fc \ dv = fcv. 

(31) .*. I fc dv = /c \ dv. 

XX I II. A constant factor may be written either before or after 
the integral sign. 

The chief application of XXIII is to be found in cases like the 
following: 



72 INTEGRATION 

To work out xxdx. If we multiply xc:lx by 2, we have an exact 

differential, since 

d(x^) = 2xdx, 

.*. \ 2xdx = x'^ ; 

but by XXIII, i 2 xdx =21 ar^;ir, 

.-. jxdx=^. 

From (31) we may also write 

(32) ff(x) dx=-^ /^/W dx. 

Integral of a Stem of Differential Expressions, If u and v 
are functions of ;r, then 

^(^/ + "t/) = — (it -V v) dx =^ du + dv, 
dx 

.'. 1 {du + dv) = I <^(?/ + v)= 7^ + V = \ d?i+ \ dv. 

This result gives Rule 

XXIV. The integral of any algebraic sum of differential ex- 
pressions equals the same algebraic sum of the integrals of these 
expressions taken separately. 

That is, e.g., 

\ (x-h 3) dx= \ (xdx + 2^^) = \xdx +\;^dx=lx'^-{-3x-\- C 

31. From any result in differentiation may always be 
derived an integration formula, and we now proceed to 
obtain some of the simpler ones, making use of § 18. 



INTEGRATION 73 

Since by VIII, 

then, integrating, 

^m+i ==C(jn + i)v'"dv = (jn+i) Cv"' dv. (XXIII) 

v'^dv^ ■ 

m+ I 



From IX, d log^ ^ = — , 

(34) ••• /-^ = loga^- 

In the same way we might go through with each formula 
in § 18. It will suffice for our purpose to tabulate a few 
of the results : 

XXV. iv^dv^^^^^^^C{jn^-V). 
J m + 1 ^ ^ ^ 

XXVI. (^ = l0geV+C. 

J V 

XXVII. (avdv = -^^-{^C. 

J loge a 

XXVIII. fsin vdv=-(io^v + C. 
XXIX. Tcos vdv = ^mv ^^ C. 
XXX. r — " ^ = arcsin-+C. 

XXXI. r /^^^ = i arc tan ^ + C. 



74 INTEGRATION 



1. Find 



EXAMPLES 

J dx 
vr^x 



This is the same thing as j (i — x)"^ dx, which resembles XXV. 

For put 

I — X = V, then — dx — dv, or dx — ~ dv. 



and by substituting again, 



. \{} — ^') '^dx — \v ^ — dv = — \v ^dv. 



.-. by XXV f f ^dv = '^ + C, 



r dx 

*^ Vi — X 



; = — 2 Vl — X + C 



2. Workout r_3^xdx^ 



U'x'^ 
Taking out constant factor 3 a (XXIII), this becomes 

xdx 



3 

and this resembles XXVI. 

For put 6'- — W'x'^ = 7/, .-. — 2 b'^xdx = dv, or xdx = -. 

2 d^ 
_ dv^ 

... 3^f_^^^ = 3^r_^'=_3_i?r^^_3^1og^ + C 
•^ J c^-b'x-^ ^ J V 2d'-J V 2d'^ "" 

Jc^-d'x^' 2d-' ^^ ^ 

3. Find r ^^ . 

J 9 + 4^^ 

This resembles XXXI, if <3: = 3, 2 x = v. 

Then 2 dx — dv, and since the given integral by (32) is the same as 



2 J 32+(2;f)2 ^^ 2J ^•'2 + 7^2' 

we find by XXXI, f ^^ = i arc tan — + C. 
^ 9 + 4^2 5 3 



INTEGRATION 75 

By studying the above examples the student will see 
that integration depends upon comparison of the given 
integral with certain standard forms. To be able to tell 
quickly what form the given integral resembles is abso- 
lutely essential. 

Tables of standard forms * have been constructed con- 
taining all integrals occurring in ordinary work. 

EXERCISE 16 

1. Prove the following integrations : 

(a) J (ax + ^;t-2) cfx=iax'' + l bx^ -f C. (Use XXIV.) 

(b) r^!£^f^ = log,(i -cos;r) + C. 
J I - cos;r 

(0 J y/aP' - x'^xdx = - J(^2 _ _;^2)l _^ c. (Use XXV, v = a^ - x^j. 

(d) \ sin {2 x)dx= — \ cos 2x -\- C, 

C r X dx 

(e) \e-dx^-er--\- C, (g) \ -;== = Va^-^x^-^ C. 

if) f— ^=='arcsin(2^) + C. (//) fJ^ = _log.(i -;r) + C 

J sin X 
tan ;r^;ir = - loge COS ;r + C. (Put tan ;r =r and use 
cos X 
XXVI.) 

(k) ism^xdx = Ix- lsin2x -\- C. (Put sin^jr = |(i - cos 2;i').) 

2. Special Devices in hitegration. 

{a) By partial fractions, when we have to integrate a rational frac- 
tion times dx^ and this fraction can be replaced by partial fiactions. 

* E.g., B. O. Peirce's A Short Table of Integrals, Ginn c^ Co., 1899. 



^e INTEGRATION 

For example, l —j—_ — 2' 

T. . I A ^ B 
Putting —^ 5 = 1 ■ — ^ 

and clearing of fractions, 

\=.x(^A- B)-\-a(^A-\- B). 
.-. A- B = o, a{A ■\-B)=i, or A=B =—- 
C dx \ C dx ^ \ C dx I /I / , X 1 / NX 

7. a ^\a-xj 
(d) By change of variable. 
Find j V^2 _ x'^dx. Substitute x = a cos 6 ; 



.-. ^;ir = — ^ sin ^ dO^ V^^ — x'l — ^ aP- — a""- cos^ ^ = ^ sin 6, 
and J V^2 _ ;^.2^;^ ^ _ a'^^sm^ede = --(9 + -sin2^+ C 

by Ex. I (/^). Now 

jjf / x'^ X 

= arc cos-, sin 2 ^ = 2 sin ^ cos ^ = 2\/i ^ • -• 

a \ a^ a 

.-. t V^2 _ ^2 cix = arc cos - + - xy/a"^ — x\ 

J 2 a 2 

3. Prove f--^!- = log,A7-^— + C. 

The following two examples illustrate the manner of determination of 
the constant of integration by means of so-called initial conditions. 

4. Find the amount of a sum of money increasing continuously at 
compound interest of r per cent. 

We found, page 43, that, in derivatives, P being the sum sought, 

dt 100 
Multiplying by dt and dividing by P., we have 

^-^-^JL-dt, 
P 100 ' 

integrating, ( i ) log^ ^ = 7^ ^ + ^- 



INTEGRATION 



77 



Let now a equal the initial sum of money ; that is, the sum started 
with, so that P — a when t — q\ substituting these in this equation, 

we have \og^a - C^ so that (i) becomes Xog^P-- — /-floge^, or. 



transposmg. 



\og.P- 



loge^ 



r ( P\ r 

100 \a I 100 ' 



t,e. 



P = ae^' 



Ans. 



5. Find the relation between s (space) and / (time) for uniformly 
accelerated rectilinear motion. 

Since the acceleration — ^ is constant, say/, we have — ^ =/. 

at at 

Multiplying by dt^ dv =fdt, and integrating, ?y =// + C. 
To determine C, let the initial velocity be v^^ i.e. v — v^ when / = o, 
or 7/q = o -f C. .'. V =ft + Vq. 

Since v =—., .-. —=ft-\-v^, and multiplying by dt, ds^ftdt-^-v^dt. 

dt dt 
Integrating, s = \ft'- -f v^t -f C, and if i" = j'q when / = o, we have 
finally s = i/t'^ + v^t + s^. Ans. 

32. Definite Integral. We have already seen that the 
indefinite integral contains an arbitrary constant, t/te con- 
stant of integration, and has for that reason an indefinite 
value. By making suitable assumptions, now to be ex- 
plained, we are able to dispose of this inconvenience. 
In § 28, Example 2, it was shown that the differential of 

the area u between a 
curve MABC, the axis 
XX\ and any two ordi- 
nates was given by 
du =:y dx. 

,'. u=^ \ y dx + C. 

Here, of course, y is 
some function of x determined from the equation of the 

curve, and .*. i ydx= some function of x, say F(x). 

.-. u = F{x)+C. 



Y 


^^ 


i 1 


f 







:i i 


J X 



Fig. 23 



78 INTEGRATION 

Let us now agree to reckon the area from the axis VV\ 
so that when x = a, // = area OaAAT, etc. 

Under this assumption, when ;r = o, ic — o, and 

.-. o=/^(o)+C, or C:=-/^(o), 

and we have 

li = F{pc) — F{o), 

Now area OaAM = F{a) - /^(o). 

Area (9^^il/ = Fib) - F{o). Subtracting, we have 

Art^ abAB =F{b)-F{a\ 
or, 

The difference of values of the 1 y dx for x = b and x = a 

gives the area bounded by the curve ivJiose ordinate is y^ the 
axis XX\ and the ordinate s at a and b. 

This difference is represented by the symbol 

(35) X-^"^-^^ 

read, ** integral from aX,o b oi y dx'' ; the operation is called 
integration between limits, a being the lozver, b the tipper 
Umit. 

We see therefore that (35) or, what is the same thing, 

(36) SlA^)d^ 

always has a definite value, and is accordingly a definite 
integral. For if 

(37) ^f{x)dx = F{x)^C, then 

(38) £f{x)dx = F{b) + C- {F(a) + C) = F(d) - F{a), 
and the constant of integration has disappeared. 



INTEGRATION 79 

33. Areas of Plane Curves. From § 32, we have the 
theorem : Given any plane curve y =/"(.r), tlie definite 

integral 1 f(x)dx gives the area boimded by that curve ^ 

the axis XX' and the ordinates at a and b. 

To find the area bounded by two given curves, we get 
the area between each and XX' and then subtract. 

Volumes of solids of revolution. 

Precisely as in § 32 and remembering the result of 
Example 3, § 28 we prove that : 

Given any plane curve y =f{x\ the definite integral 

j iry'^dx gives the volume generated by revolving around 

XX' the portion of the curve between the ordinates at a 
and b. 

The two theorems just given find numerous applications 
in Geometry. 

EXERCISE 17 

1. Find the area of the curve / — jr-^ — 9 lying below XX' , 

Here \ ydx= \ (x^— 9) cix, and since for / = o, ;r = ± 3, the limits 
are +3 and —3, z.e. area = \ (x'^—g)dx. 36. A;/s. 

2. Find the area of the circle x^ + y"^ = a'^. 

Since / = Va- —x-y \y dx= \ \/a-— x-dx which has been worked 

out in Exercise 16, Example 2 (^). For the semicircle the limits 
are + a and — a. 

3. Show that the area of the ellipse b-x'^-{- a^y'^ = ^2^- is to the area 
of the circle whose diameter is the major axis 2 a 2iS b \ a. 

4. Find the area of one arch of sine curve y — sin x. 2. Ans. 

5. Find area between the equilateral hyperbola xy — \. the axis 
A'^', and the ordinates at ;ir = ^, ;r= <^. logef-)* ^^^^- 



8o 



INTEGRATION 



6. Find the volume of the sphere. 

Since we have to revolve the circle x^-\-y^ = a% or y^ = a^-'X^ 

around XX' y then J iry'^dx = ttJ (a^ -x^) dx. The limits are + a and 
— a. ^ira^. Ans. 

7. Find the volume generated by revolving around XX' the pa- 
rabola /^ —^^^ and cut off by a plane perpendicular to XX' at the 
distance of 4 to the right of the origin. 32 tt. Ans. 

34. Definite Integral as the Limit of a Sum of Differential 
Expressions. In the Differential Calculus the student was 
asked to bear in mind that everything was built up from a 
fundamental limit, the limit of a quotient whose de7toniinator 
approached zero. We are now to see that the definite inte- 
gral is the limit of a sum of dijfe^^ential expressions. 



If 



^f{x)dx^F{x)^-C, 



then 4- ^W =/(^) and V f{x)dx =F{b) - F{d) 
dx ^^ 

gives the area bounded by the curve j =/(;ir) (Fig. 24), 

the axis XX^ , and the ordinates 2X x^ a, x = b. 



^— i? 



i>i 



P.r-^ 



-n^ 



^. 



Ps 



a a?j bi X2 &a a?3 b^ x^ b^ x^ 65 

Fig. 24 



a?6 b 



Now divide the segment ad into any number of equal 
parts, say 6, a-^d^ = 3^^^= ••• == b^d, and call the length of 
each division Ax, Erect the ordinates at these points, and 



INTEGRATION 8 1 

apply the theorem of the mean (§ 26) to each division. 
In the present case F{^) takes the place of /(;r) in (21), 
and /{x) replaces /'(x) ; for the first interval a^^y a = a, 
d=d-^, and;iri, lying between a and d, is marked in the figure. 
Draw the ordinate of x^. Then (21) gives 

or, since b-^^ — a = Ax, 

(39) F{b,)-Fia)=f(x,)Ax. 

In the same way (21) applied to each of the remaining 
five segments gives the equations 

F{b,)-F{b,)=/{x,)Ax, 
F{b,)-F{b,) = /{x,)Ax, 

(40) \F{b,)-F{b,)=/{x,)Ax, 
F{b,)-F{b,)=f(x,)Ax, 

{F{b) -F{b,)=/{x,)Ax. 

Adding the six equations (39) and (40), we find 

(41) F{b) - F[a) =f{x^)Ax +f{x^)Ax +f{x^)Ax 

+f{x^)Ax +f{x,)Ax +f{x^)Ax. 

But f{x-^Ax = area of the rectangle aPP-Jj^, 
f{x^)Ax = area of the rectangle b-J)-^P^b^, 
etc., 
so that the sum on the right equals the area 

aPP^p^P^p^P^p^P^PiP.P.Qb ; i.e. 

(42) F{b) —F{a) = area between the broken line 

PP,P,-P,Q^ridXX', 

EL. CALC. — 6 



82 INTEGRATION 

and this is true independently of the number of parts into 
which ab is divided. Hence for any number n of equal 
parts 

(43) F{b)^F{a) =/(^-i)A^ + Ax^)l^x + ... 4-/(;r,)A;r, 

(44) and A;r = ~ ^ ' 

n 

Equations (43) and (44) hold when ;/ increases without 
limit, and then Lx becomes dx {\ 27), i.e. a variable whose 
limit is zero. 

.-. F{b)-F{a)= L™^ U{x^)dx+f{x^)dx+ ... +/(;r„)^), 

or, by (38), 

(45) £Axy^ = n'^t {f{x,)dx+f{x^)dx+ ...+f{x„)dx). 

And now we see very clearly why I f{x)dx gives the 
area under the curve, for as n increases, the broken line 
PP1P1P2P2 '''PbQ approaches the curve itself, and the sum 

f{x^dx-\ h/^trj^/.r always represents the area under 

this broken line. 

Integrating between hmits is accordingly spoken of as 

"summing up"; the integration sign J is historically a 
distorted 5, the first letter of siun. But let the student 
not forget that the definite integral is not a sum, but the 
hfuit of a sum, the number of terms iiicreasing without 
limit, and each term itself diminishing toward zero. 

The problem of finding the area is then to be thought 
of thus : Divide the interval on xx^ into any number of 
equal parts, and at a point within each division erect an 
ordinate to the curve; construct the rectangles on the 
divisions as bases, with the corresponding ordinate as 



INTEGRATION 83 

altitude. Then finding the area consists in summing up 
these rectangles and taking the limit of this sum as the 
number of divisions increases without limit. 

As an example of the great number of problems in Physics and other 
branches of Mathematics which involve in their solution definite inte- 
grals, consider the following : 

To determine the amount of attraction exerted by a thin, straight, 
homogeneous rod of uniform thickness and of length / upon a material 
point F of mass ;//, situated in the line of direction of the rod. 



dx^ 



Fig. 25 

Imagine the rod (see Fig. 25) divided up into equal infinitesimal 
portions (elements) of length dx. If M = mass of rod, then 

— dx= mass of any element. 

The law of attraction being Newton's Law. i.e. attraction = product 
of masses -f- square of distance, then 



attraction of element dx on P -. 



^-mdx 



{x^-ay 
and the total attraction is the s?i?rt of these from ,r = o to :r = /. 

— ;;/ dx 

.-. Force = Cl = ^ C ^^ , 

.1. (x+aY I Jo {x-^ ay 

or integrating. Force = ^^ ( — + -] = MUL^, Answer. 



CHAPTER V 

PARTIAL DERIVATIVES 

35. Functions of More than One Variable. In the pre- 
ceding chapters we have been concerned with functions of 
one variable; i.e, the variable function depended for its 
value upon the value of a single variable. Such functions 
do not by any means suffice for the applications of the 
Calculus. In fact, the student is already familiar with 
many examples of a variable whose value depends upon 
those assigned to two or more distinct variables. Thus 
the area of a rectangle is a function of tzvo variables, viz. 
the two sides ; the volume of a gas depends upon both the 
pressure and the temperature ; the volume of a parallele- 
piped depends upon the three edges, etc. 

Notation, If the value of a variable u depends upon 
two variables, x and y, and can be computed when values 
are assumed for x and y, then we write precisely as in § 3, 

(46) u^f{x,y\ 

Similarly for a function of three variables, 

u = ^{x, y, z\ etc. 

36. Partial Differentiation. As in § 12 the important 
question arising here is how to determine the manner of 
variation of the function when the variables change in 
value. But we have greater latitude here than in § 12. 
For in (46) we can ask ourselves, 

84 



PARTIAL DERIVATIVES 85 

firsts how does it vary when x alone varies and y remains 
constant ? or 

second^ how does u change when x remains constant and 
y varies ? or 

thirds in what manner does u vary when both x and y 
change i7idependently of each other ? 

Thus let u = xy^ X and y being respectively the base and altitude 
of a rectangle; if/ remains constant (say y — b). ti gives the area of 
all rectangles of a certain altitude b ; and \i x =2, constant, say a^ then 
u represents the area of all rectangles with common base a. But if 
X and / both vary independently, then we are to consider all possible 
rectangles. 

Now the first and second cases do not differ in the least 
from § 12, for we really have in \\\^ first, ti a function of x 
alone, and in the second^ tc a function of y alone. We can 
therefore form, 

first, the increment quotient (§ 13) when x alone varies, 
and this is 

^^^^ Ax A^ ' ^^ 

second, the increment quotient when y alone varies, which 
is 

(48) A^/_ /(^,j/ + Aj/)-/(^, j) 

^ Ay Ay ' 

For example, in the area of rectangle already used, u = xy, 

t^u _ (r + Ax^y - xy , A/^ ^ 

A^- ^ =/, and ^- reduces to -r. 

Finally, we can, as in § 14, find the limits of the func- 
tions in the right-hand members of (47) and (48), in (47) 
when Ax approaches zero, in (48) when Ay approaches 



86 PARTIAL DERIVATIVES 

zero. The results are called the partial derivatives of u 
or f(x^ j) with respect to x and y respectively, and this 
step of passing to the hmit we indicate on the left by 
replacing the A's by roimd (9's, so that 

The partial derivatives — , — are then to be calculated 

bx by 

by the rules of Chapter II, the independent variables being 
respectively x and y, 

EXERCISE 18 

1. Find the partial derivatives of: 

* \:r/ bx X by y 

(2) u = .rctzn(l). Ans. |^ = -^^; f-' = 

\xl bx x^ + y^ by 



^2 4- jK^ by x'^ + y 

(3) ;/ = xv. Ans. ^ =yxy-^ ; ^ = xy log, x. 

^^^ bx by 

Partial Differentials : 

By § 27, (29), the differential of tc, when ;ir alone varies is 
- dx, and when y alone varies e* 
called the partial differentials of u. 

(so 



— dx, and when y alone varies equals — dy; these are 
Bx -^ ^ dy ^ 



bu 

— dx^ partial differential of ;/, when x alone varies ; 
bx 

— dy = partial differential of u, when y alone varies. 
by 



PARTIAL DERIVATIVES 87 

37. Total Differentiation. We have yet to discuss the third case 
of § 36, viz. required the change in u when x and/ vary independently. 
If A^, A/, and A?^ are the increments of these variables, then from (46) 
we have 
(52) t^u =f{x + Ar, / + A/) -f{x, y). 

By adding and subtracting /(;t', y -|- A/) in the right-hand member, 

(52) becomes 

(53) A/^=/(^+A^', / + A/)-/(-^. / + A/)+/(;r, yJr^y)-fix, y). 
Consider now the last two terms, 

/Gr,/ + A/) -f{x,y). 

This is the increment of ?^ or f{x^y) when y alone varies. Hence, 
by (27), § 27, 

(tA) /C-^'?/ + Ay) ~/(-^*?/) = — ^y -T terms in higher powers of A^'. 

In the same way the first two terms of (53) give us, if we set 
u' =/(,r,/ + Aj), 

(55) /(^+ A^r/ + A/) -/(--^^z + Ay) 

= — A;t' + terms involving Kx^, etc. 
dx ^ 

But also 

u' =f(x,y + A/) =/(^,/) 4- ^^A/ + terms in ^^ etc., 

ay 

by (26), § 27. Differentiating with respect to x, we find 

(56) ^ = 17-+-'^^^'^^^' 

since ^ =/C-^''7)- 

Consequently, from (56), (55), and (54). (53) becomes 

(cn\ ^u = — A.r + —A/ + terms of higher degree in Ar, A/. 
^ -^ '^ ^ (9^r 6*/ 

Now letting Ar and \v approach zero, i.e. become the infinitesimals 
dx and dy. then, as in § 27, calHng the prificipal part of A// the total 
differential of //, we have 

(58) ""%■"-%■"■ 



88 PARTIAL DERIVATIVES 

From (51) and (58), then, we have the theorem : 

The total differential of a fimction of several variables 
equals the sum of the partial differentials. 

Example. In § 28, Example i, was demonstrated the result 

d (xy) = xdy + y dx, 
which agrees with (58). 

EXERCISE 19 

Find the total differentials of the following : 

y dx — xdy 



{a) ^ = loge^-j. 
{b) u — arctanf- j. 



A71S. 


du = 




xy 


Ans. 


du = 


xdy 
^2 


— y dx 

4- v2 • 



{c) u = xy. Ans. du = xy-^{y dx ■-{- xXogeXdy), 

38. Total Derivative. We may in (57) assume that x 
and y are not independent, but are functions one of the 
other, say J/ a function of x. Then u becomes also a func- 
tion of X alone, and we may therefore form the total 

derivative — 
dx 

Dividing (57) by l^x and taking the limit for l^x = o, and 

.'. A7 = o, we have the result 

dx dx \dyj dx 
a very important formula. 

Suppose in the illustration of the rectangle, § 36, we wish the deriva- 
tive with respect to the base x of the area u of all rectangles whose 
altitude/ is double the base. Then 

u^xy, y = 2x, ^=j, ^ = x, ^ = 2, 
^' ^ ' dx -^' dy dx ' 

and (59) gives —=y-{-2x=4x, 

dx 



PARTIAL DERIVATIVES 89 

Or, we may substitute for 7 before differentiation ; 

i,e, u = X ' 2 X = 2 x'^j .', — = 4 :r, as before. 

ax 

Equation (59) is especially important as affording a proof of the 
method given in § 19. For in the example of that article, set 

u =1 x^ — 2, xy •{■ 2/2 _ 3 . 

.-. « = o, and — = o, or _ + _ -=i = o ; 
ax dx \dy)dx 

du 

U. (60) ^:=-|f=- ^"-3/ ^ll^^l, 

^ ' dx du — 2X+4j'2x-4y 

the same answer as before. This formula (60) is very useful. 
For further study of the Calculus the student is referred to : 

G. A. Gibson, An Eleinentary Treatise on the Calculus. London, 

1901. 
Young and Linebarger, The Elements of the Differential and 

Integral Calculus. New York, 1900. 
McMahon and Snyder, Elements of the Differential Calculus. 

New York, 1898. 
Murray. An Elementary Course in the Integral Calculus. New 

York, 1898. 



CHAPTER VI 

ADDITIONAL EXAMPLES IN CONNECTION WITH 
THE GIVEN EXERCISES 

EXERCISE 1. PAGE 10 

6. Given f{x) =x^ — 6 x'^ -\- ^\ 
then by § 3, /(2) - (2)3 - 6(2)2 + 3 

= 8 - 24 + 5 = - II. 
Similarly, /(- 3) = (- 3)^ - 6(- 3)2 + 5 

= - 27 - 54 + 5 = - 76. 
Also f{x - i) = (;r - 1)3 - 6(;r- - 1)2 + 5, 

which reduces to =;i3 — 9 .r2 -f 15 ;ir — 2. 

7. Given ^(^) = 3^^ — 2;ir2 + 6;r, prove 

i^(o) = o; /^(i)=7; /^(-2) = -44; ^(-f) = -^F; 

8. Given <^(;r) ^ "^"^ \ prove the following: 

9. In Example 7, prove <^[^^- )=/• 

EXERCISE 2. PAGE 22 

7. Prove the following : 

Limit rcsc;r1 =00; Limit T^-^^l =co; Limit f'^-^'^l = i- 
L Jx=o L:r~iJ^i L;ir-iJ^=oo 

90 



6, Draw the graphs of 5^ ; — ; Vi - x^; csc;f; cot;»r. 



ADDITIONAL EXAMPLES 91 

EXERCISE 3. PAGE 24 

It is often convenient to place the variable / equal to the given func- 
tion of X. Then when x changes to ;r + //, y is replaced by / + A/, 
that is, A_y is the increment of the function. The following examples 
illustrate this, and in them the increment of x is denoted by /\x instead 
of hj as heretofore. 

9. Given y = x'^ — 2fX-\-6\ find A/. 

Substituting x + A;r and / + A/ for x and / respectively, we have 
J + A/ = (;r + A;r)2 _ 3(;tr + A;i-) + 6 

= x'^ -\- 2 X ' Lx -\- (A^)2 — 3 X — 3 A;r 4- 6. 
Subtracting from this the original equation, we find 
^y = 2 X' A;r + (A;r)2 — 3 A;f, 
or Ay =(2 X— ;^ + Ax) Ax. Ans. 

10. Given y == x^-^ find Ay. 

A/ = (3 ;r2 + 3 r . A;i- + (Axy)Ax. Ans. 

11. Given / = — — — : find Ay. 

X 

. x^ — I + X ' Ax . M 

Ay — — Ax. Ans. 

x'^ -\- X ■ Ax 

12. Given _y = -^ ; find Ay. 



Ay = = :::::^:::::^:3 Ax. Ans, 

(x + A;r) Vx + x\/x + Ax 



EXERCISE 4. PAGES 32-33 

As above, if y is set equal to the given function of x to be differ- 
entiated, the General Rule of Differentiation (page 29) may be stated 
as follows in Four Steps : 

1st step. Snbstitnte x + Ax and y ^ Ay for x and y^ respectively. 
2d step. Subtract the original equation from the new equation, and 

reduce. 
3d step. Divide both sides of this equation by Ax. 
4th step. Find the limit of this result when Ax approaches the limit 



, . Ay . dy 
zero, replacing -^ by ^^. 



92 ADDITIONAL EXAMPLES 

9. Differentiate y = ;^x^ + 2x-\-^. 
Carrying out the Four Steps : 
1°. Substituting x + Ax and / + Ay for x and/, 

/ + Ay = 3(^ + A;r)2 ^ 2(x -\- Ax)-\- 5, 
or y-\-Ay = ;^x^-{-6x' Ax + 3 (Axy -\- 2 x -{- 2 Ax + 5. 

2°. Subtracting the original equation and factoring, we have 

Ay =(6x+ ^Ax-\- 2) A;r. 
3°. Dividing by Axy 

Ay ^ 

^=:6;r+3A^+2. 
4^ Letting Ax approach the limit zero and replacing -^ by ^> 







-f- = 6 AT + 2. Ans. 
dx 


10. Differentiate 


X 

-^ ~ I + ;r2" 


Working 


out the 


Four Steps gives : 


T° 


/ + 


Av- ^^^"^ 


1 • 


^ i+(,r+Ax)2 


1*^ 




;f + A;i:- X 
At/ — -■■ • 


ji • 


■^ l+(;r+A;r)2 i4.;^2» 


reducing, 




I — ;r2 — :r • A;t- 


-^ (I +;t-2)(i +(;r4- A;r)2) 


3°. 




A/ I — ;i-2 — ;f . A;ir 
A:r~(i +;f2)(i +(;t-+ A;r)2) 


4^ 




^/ I - ;r2 


11. Differentiate 


/=i^'-^2-3^+5- 






-^ = :r2 - 2 ;r - 3. ^;/j'. 
dx ^ 



Ax. 



12. Find the slope of the tangent to the curve y = ^x^ — x^ — ^ x-\- ^ 
at the points where x = o, ^=3, x = — i, x— i. 

From Example 1 1 and page 27, the general expression for the slope 

at any point is given by the value of -j-, i.e, 

slope = :r2 - 2 ;jr - 3. 



ADDITIONAL EXAMPLES 93 

Hence, substituting, 

slope at ;f = o is — 3 ; 
slope at ;r=3 is 9-6 — 3 = 0; 
slope at;r=— lis i +2 — 3 = 0; 
slope at ;r = I is I — 2 — 3 = — 4. 

13. Find the general expression for the slope of the tangent to each 
of the following curves ; 

(a) y = x^ - Sx^ + 16. 4X^ - 16 x. Ans, 

... 2X 8-2;i'2 

^ ^ -^ 4 + :r2 (4 4- x^y 

_ 8 ^^ — 16 a^x 

14. Determine the coordinates of all points on each of the curves of 
Example 13 at which the tangent is parallel to the axis of x. 

Hint. Since at these points the slope of the tangent is zero, place 
the general expression for the slope equal to zero, and solve for x. 
(a) (o, 16), (±2,0); (d) (2, 1), (-2, -i); (c) (o, 2 a). Ans. 

EXERCISE 8. PAGE 48 

c^y X — y 

8. Given x^ — 2 xy = a'^-^ prove -^ = ~ - 

9. Given (x - ay + (/ - dy = r^; prove ^^ = -^"^ ^. 



i^x y — b 
ill 

dx~ ^x 



10. Given x^ -{■ y^ = cfi \ prove -^ = — ^\^. 



11. Given r^ sin 2 ^ = ^2 . prove -^ = — r cot 2 ft 

dd 

12. Given the curve /^ = x ; show that the axis of^' is the tangent at 
the origin. 

EXERCISE 9. PAGE 49 

If the variable/ is set equal to the function of ;i' to be differentiated, 
then the first differentiation gives the value of ---- The second de- 
rivative, obtained by differentiation from this, is represented, as in 
§ 20, by ^^ (read, "the second derivative of / with respect to ;r"). 



94 ADDITIONAL EXAMPLES 

Then further successive differentiation will give the third, fourth, fifth 

d^y d^y d^y 
derivative, and so on, which are symbolically -— , -^, -^, etc. 

d^y 

10. Given / = ^~^ cos x ; prove -j^ = — 4/. 

d^y 2 X 

11. Given J = arc tan ;r ; prove -7^ = — - 



dx^ (1+ x-y 

12. Given/ = loge (i ■\- x^y -, prove ^= ^1-Z^^. 

d^y 

13. Given y = x cos :r ; prove -t\ — :r cos r -f 4 sin x, 

14. Given j = sin^ 2 ;ir ; prove -7^ = 8 cos 4 x, 

15. Given ;r- + 2 ;ry = ^^ ; prove -7^ = — 5- 

d^y 

16. Given j = tan- x -\- S log cos ;r + 3 ^^ ; prove --t4 = 6 tan^ x. 

17. Givenj = ^— ^; prove ^ = -^^^—^. 

18. Given j/ = (x^ + rt:^) arc tan - ; prove — 



19. Given y^ -\- y = x^ ; prove 



a' ^ dx^~ (a^ + x^y 

d^y — 24X 
dx^~ (I + 2y)^ 



20. Given y^ — 2 m xy -\- x^ - a = o : prove Vr, = -7^ t4- 

"^ ^ ^^--^ (/ — w;r)^ 

21. Given y = ^ ; prove ^ = -— ^^ tt^- 



EXERCISE 10. PAGE 52 

5. Find equations of tangents and normals to the curve /^ = 
2 x^ — x^ 2Lt X = I, Ans. Tangents 2iXt y — \x -\- \\ y — — \ x — \. 

Normals are j = — 2;i*+3; j = 2;ir— 3. 

6. Prove that subtangent and subnormal to the ellipse b'^x'^ + a^y'^ 

= ^2^2 at (^x\ y') are ^ — and ^, respectively. (Use formulae 

of Example 3.) 



ADDITIONAL EXAMPLES 95 

7. Find equations of tangents and normals to 2 x^ -\- ^y^ = ^^ 
at / = 3. Ans. Tangents are 4 ;r± 9/ = 35. 

Normals are 9jrT4/ = 6. 

8. Find equations of tangent and normal to jj/ = x^ at (x', y'). 

Ans. Tangent is 3 x"^x — y — 2y' = o. 

Normal is ;t- + 3 x''^y — x\^ x"^ -\- i) = o, 

X 

9. Find equations of tangents and normals to jk = ^ 3,t x = i . 

Ans. Tangents are y = ± I. 
Normals are x = ± i. 

10. Prove that the equation of the tangent to x^ — 3 axy -\- y^ = o 
at (x\ y) may be written x''^x — ax'y — ay'x -\- y'^y = o. 



EXERCISE 11. PAGE 54 

4. Prove that the curve/ = — has points of inflection at :r= o 

and:r = ±^V3. a -j- x^ 

5. Examine the curve a^y = ^x^ — ax^ + 2 a^ for points of inflection. 

Ans. (a, I a). 

6. Examine y = x -{- ^6 x^ — 2 x^ — x"^ for points of inflection. 

Ans. At ;r = 2, ;ir = — 3. 



EXERCISE 12. PAGE 57 

Referring to Fig. 14, page 55, we see that the conditions for maximum 
and minimum values of a function may be stated as follows, the graph 
of the function being supposed drawn : 

Maxinmni: tangent horizontal and graph below the tangent (e.g. at P^) ; 
Minimum : tangent horizontal and graph above the tangent {e.g. at P^ . 

If we set / equal to the given function of ;r, then the preceding may 
be stated thus : 

Maximu77t : -^- — o and --^< o ; 
dx dx'^ 

Mifiimum ; -^ = o and — 4> o. 
dx dx^ 



96 ADDITIONAL EXAMPLES 

This is identical with the Second Test, page 56, and the process is 
summarized in the following rule : 

1st step. Set y equal to the giren function of x and work out 

expressions for ^ and ^. 

2d step. Set the expression found for ^ equal to zero, and solre 
^ {too 

for X. 

3d step. Substitute each yalue of x thus found in the expression 

for 3-^« If this result is negative, the corresponding 

yalue of 1/ is a maximum ; if positive, the correspond- 
ing value of 2/ is a minimum. 

To illustrate, solve the problem : To find the maximum and minimum 

values of 

2 X 

Following the rule, we have 

X^ -\- A 

i^ Assume J = -; then, differentiating and reducing, 

4y _^^ - 4 , 

dx^ 2X^ 

dx^ x^' 

2^ Setting the expression for -— equal to zero gives, 

;r2 - 4 = o, 

or x = ±2. 

d^y A I 
3°. If ^=2, then af^ = Jz = r 

dy 4 I 

if ^=-2, then ^2 = 3T3 = -r 

Therefore, since 

x= 2 gives y = 2f 

and x= — 2 gives y = — 2, 

then 2 is a minimum value, and --2 a maxifnum value of the given 
function. 



ADDITIONAL EXAMPLES 97 

4. Find maximum and minimum ordinates in each of the following 
curves, and draw the loci : 

{a) y = 2x^ — ^x^ — i2x-\- 4. 

Ans. Max. value 11, min. value — 16 

(Jji) y = 2 x^ — 21 x^ -{• ^6 X — 20. 

Ans. Max. value — 3, min. value — 128 

(c) y = x^ - 2 x'^ + 10. Ans. Max. value 10, min. values ± 9 

(^) y = (^x — iy{x — 2)2. Ans. Max. value 3^1%, min. value o, 

(e^ y = l^Sjf. Ans. Max. value - 

^ "^ X e 

(y) / = sin 2 ;f — ;ir. Ans, Max. values when x — {n •\- ^)7r ; 

Min. values when x — {n — |)7r, 

n being any integer. 

(cr\ y = ^^^^ — Ans. Max. value J V2. 

^^^ -^ i+tan;r 

(^n) / = sin ;ir(i + cos x). 

Ans, Max. value IV3, min. value — |v^- 

(/) y z= x^ — 4. Ans. Min. value — 4. 

(y) y = x^ — S, Ans. No max. nor min. values. 

(^) J = 4 — :r^ ^^/i". Max. value 4. 

EXERCISE 13. PAGE 58 

8. Find the altitude of the cone of ma;cimum volume which can be 
inscribed in a sphere of radius r. A 71s. -J r, 

9. Show that the radius of the base of the cylinder of greatest 
lateral surface which can be inscribed in a sphere of radius r is ^r V2. 

10. Find the shortest distance from the point (2, i) to the parabola 
y'^ = 4.x. Ans. V2. 

11. Determine the area of the greatest rectangle which can be in- 
scribed in a given triangle whose base is 2 ^ and altitude a. Ans. I ab. 

12. If the shape of a window is a rectangle surmounted by a semi- 
circle, and if the perimeter is given, show that the maximum light is 
admitted when the heiglit and breadth of the window are equal. 

13. Determine the altitude of the minimum cone which can be 
circumscribed about a given sphere of radius r. Ans. 4r. 

EL. CALC. — 7 



98 ADDITIONAL EXAMPLES 

EXERCISE 14. PAGE 64 

Prove the following expansions : 

5. ^si"^ = I + ;r-f ^;r2 -^;f4 - ^'^jfS-l- .... 

6. arc tan x = x — 1 h •••• 

3 S 7 

x^ 5 x^ 

7. sec :r= I 4-|— - + -r— + •••• 

8. logsec^=r- + - + -+-. 

2 X^ 

9. V sec ;r = i -\- x -^ x^-\- -—- + •••. 

10. sin (i + 2 ;r) = sin I + 2 cos i . ;jf + 2 sin i * x^ ^ .... 

11. log (i + ^^) := log 2 + ^ ^ + i ;r2 . 



12. Vi -\- 4x-\- 12X' = I -it 2x-^ 4x^ -\- '". 

EXERCISE 16. PAGE 75 

Prove the following integrations : 

6. \cos'^xdx=lx-hism2x-\-c. (Put cos^jr = ^(i + cos2;r).) 

8. ^^^=x-ix^ + ix^-\og(x-i)+c. 

Hint. Divide numerator by denominator before integrating. 

9. 1(3 ax'^ + 4 ^^3) 3(2 ax-h 4 bx^) dx = 1 (3 ax^ + 4 ^;r3)l -f c. 

10. f ^^2x ^;^ ^ -^ ^2x + ^. 

J 2 log ^ 

11- 1 COS (mx -\- n)dx— — sin (jnx ■\- n)-\- c, 

12. r_£^^= iarcsini;r2 + ^. 
•^ V4 — ;r4 

13. f — arc tan (;r — i ) -f- c. 

Hint. Write denominator in the form i + (;r — i)^. 



ADDITIONAL EXAMPLES 99 



14. I —^=:=:^ = \ arc sin h ^. 

*^ V3 — 4 ;i- — 4 ;r2 2 



15. 



16. 



C x'^dx 4 , 1 ^ . X . 



EXERCISE 17. PAGE 79 

8. Find the area bounded by / = 9 — r- and the axis ^^'. Ans. 36. 

JIT 

9. Find the area bounded by / = -^ and ^^' from the origin 

to :r = 8. Alls, loge V65 = 2.087. 

10. Find the area bounded byy^ = 9 r and_y — 3 x. Ans. \. 
Hint. The area required is the difference of the areas bounded by 

the individual curves, the axis XX' ^ and the ordinates at their points of 
intersection. 

11. Find the areas bounded by the following curves. 

{a) y = x'^ — 4. and XX'. Ans. io|. 

(d) y = S + 2x^^ - x^ and XX'. Ans. 29}!. 

(^) j2 = 9;rand jr = 4. Ans. 32. 

(^d) xy = 4. and x + y = 5. Ans. 6.1 14. 

12. Find the volumes of the solids of revolution generated by revolv- 
ing around XX' the following areas : 

(a) Of Example 8. Ans. 259J tt. 

(i) Of Example II (^). Ans. 341^^5 tt. 

(c) Of Example 11 (c). Ans. 7277. 

(d) Of Example 10. Ans. f tt. 

Hint. This is given as the difference of the volumes of the two solids 
generated by the two areas mentioned in the previous Hint. 

L.cfC. 



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Part II. Classification and description of each species 
with Key. 

Part III. The study of Birds in the field, with Key for 
their identification. 

Part IV. Preparation of Bird specimens. 

The descriptions of the several species have been pre- 
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those in other books. They are short and so expressed 
that they may be recalled readily while looking at the 
bird. They are thus especially adapted for field use. The 
illustrations were drawn especially for this work. Their 
number, scientific accuracy, and careful execution add much 
to the value and interest of the book. The general Key to 
Land and Water Birds and a very full index make the 
book convenient and serviceable both for the study and 
for field work. 



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Text-Books in Geology 

By JAMES D. DANA, LL.D. 
Late Professor of Geology and Mineralogy in Yale University, 

DANA'S GEOLOGICAL STORY BRIEFLY TOLD . . . $1.15 
A new and revised edition of this popular text-book for beginners in 
the study, and for the general reader. The book has been entirely 
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In contents and dress it is an attractive volume, well suited for its use. 

DANA'S REVISED TEXT-BOOK OF GEOLOGY . . . $1.40 
Fifth Edition, Revised and Enlarged. Edited by William North 
Rice, Ph.D., LL.D., Professor of Geology in Wesleyan University. 
This is the standard text-book in geology for high school and elementary 
college work. While the general and distinctive features of the former 
work have been preserved, the book has been thoroughly revised, enlarged, 
and improved. As now published, it combines the results of the life 
experience and observation of its distinguished author with the latest 
discoveries and researches in the science. 

DANA'S MANUAL OF GEOLOGY $5.00 

Fourth Revised Edition. This great work is a complete thesaurus of 
the principles, methods, and details of the science of geology in its 
varied branches, including the formation and metamorphism of rocks, 
physiography, orogeny, and epeirogeny, biologic evolution, and paleon- 
tology. It is not only a text-book for the college student but a hand- 
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investigations and developments in the science, especially in the geology 
of North America, led to the last revision of the work, which was most 
thorough and complete. This last revision, making the work substantially 
a new book, was performed almost exclusively by Dr. Dana himself, and 
may justly be regarded as the crowning work of his life. 



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BOWNE'S THEISM 

BY BORDEN P. BOWNE 

Professor of Philosophy in Boston University 

FOR COLLEGES AND THEOLOGICAL SCHOOLS 

PR.ICE, $1.75 

THIS BOOK is a revision and extension of the author's 
previous work, " Philosophy of Theism." In the 
present volume the arguments, especially from episte- 
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has been largely rewritten, and about half as much additional 
new matter has been included. 

The author, however, still adheres to his original plan of 
giving the essential arguments, so that the reader may discern 
their true nature and be enabled to estimate their rational 
value. He does this from the conviction that the important 
thing in theistic discussion is not to make bulky collections of 
striking facts and eloquent illustrations, nor to produce learned 
catalogues of theistic writers and their works, but to clear up 
the logical principles which underlie the subject. From this 
point of view the work might rightly be called the " Logic of 
Theism." 

Special attention is given to the fact that atheistic argu- 
ment is properly no argument at all, but a set of illusions which 
inevitably spring up on the plane of sense-thought, and acquire 
plausibility with the uncritical. The author seeks to lay bare 
the root of these fallacies and to expose them in their base- 
lessness. In addition, the practical and vital nature of the 
theistic argument is emphasized, and it is shown to be not 
merely nor mainly a matter of syllogistic and academic 
inference, but one of life, action, and history. 



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Mt. Shasta — A Typical Volcano 
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Biology and Zoology 

DODGE'S INTRODUCTION TO ELEMENTARY PRACTICAL 
BIOLOGY 

A Laboratory Guide for High School and College Students. 
By Charles Wright Dodge, M.S., Professor of Biology 

in the University of Rochester $1.80 

This is a manual for laboratory work rather than a 
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of drawings the results obtained. The work consists 
essentially of a series of questions and experiments on 
the structure and physiology of common animals and 
plants typical of their kind — questions which can be 
answered only by actual investigation or by experiment. 
Directions are given for the collection of specimens, for 
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also for performing simple physiological experiments. 

ORTON'S COMPARATIVE ZOOLOGY, STRUCTURAL AND 
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By James Orton, A.M., Ph.D., late Professor of Natural 
History in Vassar College. New Edition revised by 
Charles Wright Dodge, M.S., Professor of Biology in 

the University of Rochester $1.80 

This work is designed primarily as a manual of 
instruction for use in higher schools and colleges. It 
aims to present clearly the latest established facts and 
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sists in the treatment of the whole animal kingdom as a 
unit and in the comparative study of the development and 
variations of the different species, their organs, functions, 
etc. The book has been thoroughly revised in the light 
of the most recent phases of the science, and adapted to 
the laboratory as well as to the literary method of teaching. 



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Standard Text-Books in Physics 



ROWLAND AND AMES'S ELEMENTS OF PHYSICS 

By Henry A. Rowland, Ph.D., LL.D., and Joseph 
S. Ames, Ph.D., Professors of Physics in Johns 
Hopkins University. 

Cloth, 12mo, 275 pages Price, $1.00 

This is designed to meet the requirements of high 
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AMES'S THEORY OF PHYSICS 

By Joseph S. Ames, Ph.D. 

Cloth, 8vo. 531 pages Price, $1.60 

In this text-book, for advanced classes, the aim has 
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fundamental experiments on which the science of Physics 
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theories and methods. 

AMES AND BLISS'S MANUAL OF EXPERIMENTS IN PHYSICS 

By Joseph S. Ames, Ph.D., Professor of Physics, and 
William J. A. Bliss, Ph.D., Associate in Physics, in 
Johns Hopkins University. 

Cloth, 8vo, 560 pages Price, $1.80 

A course of laboratory instruction for advanced classes, 
embodying the most improved methods of demonstration 
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suggestions as to the value and bearing of the experiments. 



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Seeley's History of Education 

By Dr. LEVI SEELEY 
Professor of Pedagogy, State Normal School, Trenton, N. J. 

Cloth, 12mo, 350 pages. Price, $1.25 

Nearly 400,000 active teachers in the United States 
are required to pass an examination in the History of 
Education. Normal schools, and colleges with pedagog- 
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and the Superintendents of Education in most states, 
counties, and cities, now expect their teachers to possess 
a knowledge of it. 

This book is not based on theory, but is the practical 
outgrowth of Dr. Seeley's own class-work after years of 
trial. It is therefore a working book, plain, comprehen- 
sive, accurate, and sufficient in itself to furnish all the 
material on the subject required by any examining board, 
or that may be demanded in a normal or college course. 

It arranges the material in such a manner as to appeal 
to the student and assist him to grasp and remember 
the subject. 

It gives a concise summary of each system discussed, 
pointing out the most important lessons. 

It lays stress upon the development of education, 
showing the steps of progress from one period to another. 

It begins the study of each educational system or 
period with an examination of the environment of the 
people, their history, geography, home conditions, etc. 

It gives a biographical sketch of the leading educators, 
and their systems of pedagogy, including those of Horace 
Mann and Herbart. 

It treats of the systems of education of Germany, 
France, England, and the United States, — bringing the 
study of education down to the present time. 

It furnishes the literature of each subject and gives an 
extensive general bibliography for reference. 



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classical Dictionaries 



HARPER'S DICTIONARY OF CLASSICAL LITERATURE AND ^ 
ANTIQUITIES 

Edited by H. T. Peck, Ph.D., Professor of the Latin Language 

and Literature in Columbia University. 

Royal Octavo, 1716 pages. Illustrated. 
One Vol. Cloth . . $6.00 Two Vols. Cloth . . $7.00 
One Vol. Half Leather . 8.00 Two Vols Half Leather . 1000 

An encyclopaedia, giving the student, in a concise and intelligible 
form, the essential facts of classical antiquity. It also indicates the 
sources whence a fuller and more critical knowledge of these subjects 
can best be obtained. The articles, which are arranged alphabetically, 
include subjects in biography, mythology, geography, history, literature, 
antiquities, language, and bibliography. The illustrations are, for the 
most part, reproductions of ancient objects. The editor in preparing 
the book has received the co-operation and active assistance of the most 
eminent American and foreign scholars. 

SMITH'S DICTIONARY OF GREEK AND ROMAN ANTIQUITIES 

Edited by William Smith, Ph.D. Revised by Charles 
Anthon, LL.D. Octavo, 1 1 33 pages. Illustrated. Sheep $4.25 
Carefully revised, giving the results of the latest researches in the 
history, philology, and antiquities of the ancients. In the work of 
revision, the American editor has had the assistance of the most dis- 
tinguished scholars and scientists. 

STUDENTS' CLASSICAL DICTIONARY 

A Dictionary of Biography, Mythology, and Geography. Abridged. 

By William Smith, D.C.L., LL.D. 

i2mo, 438 pages. Cloth $1.25 

Designed for those schools and students who are excluded from the 
use of the larger Classical Dictionary, both by its size and its price. All 
names have been inserted which one would be likely to meet with at the 
beginning of classical study. 



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MAR 171904 



